CONFIGURATIONS OF SUBLATTICES AND DIRICHLET-VORONOI CELLS OF PERIODIC POINT SETS Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) Der Fakultät für Mathematik der Technischen Universität Dortmund vorgelegt von Dipl.-Math. Marc Christian Zimmermann am 14. August 2017 Dissertation Configurations of sublattices and Dirichlet-Voronoi cells of periodic point sets Fakultät für Mathematik Technische Universität Dortmund Erstgutachter: Prof. Dr. Rudolf Scharlau Zweitgutachter: Prof. Dr. Achill Schürmann Tag der mündlichen Prüfung: 22.09.2017 Contents Contents i Introduction iii What this thesis is about. And why. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Use of computer algebra systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of notations ix I Similar sublattices and finite quadratic modules 1 1 Quadratic modules and lattices 3 1.1 Quadratic forms over rings and fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Local notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Lattices over Z and its localizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 M.t.i.s of regular quadratic modules over finite rings 11 2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Regular quadratic forms over finite rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Counting isotropic elements in regular quadratic modules over finite local p.i.r. . . . . . 19 2.4 A classification of m.t.i.s. of regular quadratic modules over finite local rings . . . . . . 23 2.5 Counting m.t.i.s. of quadratic modules over finite principal rings . . . . . . . . . . . . . 30 2.6 Low rank cases over finite principal ideal rings . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Similar sublattices 41 3.1 Similar sublattices of lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Genus-similar sublattices of lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Existence and enumeration of similar sublattices . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 A glance towards Zeta functions for counting (genus-)similar sublattices . . . . . . . . . 51 i CONTENTS II A locally explicit formula for the quantizer constant in monohedral periodic vector quantization 53 4 Geometric foreplay 55 4.1 Lattices and periodic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Point sets and tilings in Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5 Periodic vector quantization 69 5.1 The normalized second moment and the quantizer constant . . . . . . . . . . . . . . . . 69 5.2 Vector quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6 DV-cells and normalised second moments 75 6.1 Voronoi’s second reduction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.2 On the DV-cell of a monohedral periodic set . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.3 A local formula and polynomial program for the Quantizer problem . . . . . . . . . . . 82 6.4 On the lattice quantizer problem in low dimensions . . . . . . . . . . . . . . . . . . . . . 87 A Table of quantizer constants 101 B Documentation of computations 103 B.1 Method of computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 B.2 Overview of routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 B.3 Worksheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Index 107 Bibliography 109 ii Introduction “The world is full of nasty numbers...” — J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, p.56 WHAT THIS THESIS IS ABOUT. AND WHY. The study of lattices and geometric properties of point sets in Euclidean space are brought together in a vast amount of mathematical research. The more prominent problems, such as the sphere packing problem and the sphere covering problem, are both classical and still current subjects of research. But there is another geometric problem involving Euclidean lattices, which has received less attention, at least from the mathematical community: the lattice quantizer problem. We will handle lattice quantization from a purely mathematical point of view. For an detailed account of its history in information theory as well as a collection of some of the most important results up to its publication, we refer the interested reader to [GN98]. For a textbook introduction to lattice quantization, also written from an information theoretic point of view, we refer the interested reader to [Zam14]. A resource from mathematics that we wish to mention is [Hem04], a Ph.D. thesis which shows where the mathematical reception of the problem stood around 2004. Since then, not to much has happened on the mathematical side of this problem1. Lattice quantization emerged as a subject of research in its own right as a special case of general vector quantization. The underlying principle is easy: suppose you have to deal with arbitrary real numbers, produced in some way and yours to handle. Maybe these are descriptions of analog phenomena like speech, or video. In any case, it might be reasonable to ask how to efficiently get a discretized, possibly finite, description of real numbers, that albeit loses only as much “information” as is unavoidable. This is the basic question of quantization: Given a source that produces real numbers x, choose a discrete (and for practical purposes finite) subset of the real line and provide a rule to assign one of those values to a number produced by that source. Or to quote: “A quantizer [...] is a device for converting nasty 1There is one notable exception in form of [DSSV09]. There, among other things, the explicit value of the quantizer constant of certain highly symmetrical lattices is derived. iii INTRODUCTION numbers into nice numbers” (cf. Chapter 2, 3.1 in [CS98]). The easiest example one can possibly think of is to use the set Z and simply round a real number to the integer nearest to it. Now a vector quantizer is based on the idea that it might be beneficial to not quantize, or round, each real number for itself, but rather to collect, say, n of them and find a way to quantize the so obtained vector. Before we talk about how to do so, we should be allowed to ask: why do we do so? The answer is that it pays off. The idea is to assign a performance measure to a quantizer and ask for such structures that optimize performance with respect to said measure. A crucial insight is that using higher-dimensional structures allows to reduce the margin of error. We will give a short account of this for lattice quantizers later on (cf. Chapter 5). In this thesis we deal with two mathematical problems that arose from this general setup. We will discuss them in two separate parts. Similar sublattices and finite quadratic modules The main goal of this part is to enumerate similar sublattices of integral lattices. More recently several authors discussed the idea to assign multiple descriptions to each quantized vector, rather than to just take the output of the quantizer as a final result. The idea here is that if certain of these descriptions would get lost, say, after sending the information over a channel, the received subset of descriptions could be used to at least get a (good) approximation of the initial code word. While we will not include a discussion of such schemes in this thesis, this was the starting point for the investigations regarding similar sublattices: the multiple-description lattice quantization schemes all use similar sublattices in the construction of the descriptions (cf. [SVS99], [KGK00], [VSS01], [DSV02], [AS10], and [AS12]). A sublattice of a lattice L is called similar if it is the image L under an isometry followed by a scaling operation λ id. Below we depict examples for similar sublattices of the standard lattice Z2 and the Hexagonal latticeA2. The superlattice consists of all points shown (black and red), while the sublattice consists of the red points only. For Z2 we show a similar sublattice with scale factor 2, it is obtained by a 45-degree rotation and followed by 2 id: (0, 0) iv WHAT THIS THESIS IS ABOUT. AND WHY. We also show fundamental parallelotopes for the super- and the sublattice, from which one can visu- ally read off that the scale factor is in fact 2. For A2 we provide an example of a similar sublattice that is obtained by a 30-degree rotation followed by 3 id. We depict A2 with lattice basis (1, 0), (1/2, ? 3/2): (0, 0) Again, it is easy to verify the scale factor to be 3. At this point we should wonder in which way the shown sublattices are special. Could we have found similar sublattices for an arbitrary scale factor? Is there just this one, or are there more? Both questions can be answered for these special lattices, but in higher dimensions both problems become increasingly difficult. However, the extent in which the mathematical literature deals with this geometric problem is rather small. What can be found in the literature, is a criterion for the existence of similar sublattices of rational lattices (cf. [CRS99]) and some results on the number of similar sublattices of certain lattices in dimensions 2 and 4 (cf. [BSZ11], [BM99], and [BHM08]). But these works often rely on certain structural assumptions that are quite special and their methods are not likely to be applicable in higher dimensions. We use an arithmetical approach to obtain results on the number of similar sublattices (under certain conditions). At least in theory this approach provides a method to explicitly construct similar sub- lattices with the use of a computer algebra system. The main results for this part are presented in Chapter 2, where we classify and count maximal totally isotropic submodules of regular quadratic modules over finite rings, and in Chapter 3, where we relate the study of similar sublattices to the results of Chapter 2. If we consider a slightly generalized situation, namely that of sublattices of an integral lattice L that are similar to some lattice in the genus of L, rather than only to L itself, the full strength of the arith- metic approach becomes visible. For certain well-behaved classes of lattices, most prominently even unimodular lattices, our approach gives a full answer to the related counting problem: for a given scale factor, or equivalently index, how many sublattices similar to a lattice in the genus of L do exist? Given L there might be a finite number of exceptions, where we cannot predict this number, the case of scale factors not coprime to the determinant of L. However, for unimodular lattices this is without consequence. v INTRODUCTION A locally explicit formula for the quantizer constant in monohedral periodic vector quantization The main goal of this part is to systematically investigate the quantizer constant and its minimal values for monohedral periodic sets. We take a set of points in Rn, use their Dirichlet-Voronoi tiling of Rn and associate to an x P Rn the Dirichlet-Voronoi cell it is contained in2. This defines a vector quantizer. If the points form the standard lattice Zn, this is just component wise rounding: v w Here the black dots are the lattice points and we depict the Dirichlet-Voronoi cell of the origin in a darker red shade. We see that the other Dirichlet-Voronoi cells of this lattice are all translates, by lattice points, of this fundamental cell. We depict some of them in a lighter red shade. As an example for quantization take a look at the points v,w. They lie in distinct Dirichlet-Voronoi cells and v would be quantized by (0, 0) andwwould be quantized by (´1, 0). Another example, where it is still easy to see how to quantize, is the Hexagonal lattice A2. We depict it with lattice basis (1, 0), (1/2, ? 3/2): vw the above remarks hold in a literal manner. 2If x is contained in several Dirichlet-Voronoi cells, which can happen if and only if it lies on their boundary, the tie can be broken arbitrarily. vi STRUCTURE OF THE THESIS A good reason why to use lattices instead of dealing with arbitrary point sets is lies in the hardness of the closest vector problem. It surely is hard for lattices, but it is much harder for arbitrary unstructured sets. We utilize the squared Euclidean distance as a measure of performance and ask to minimize the mean error of quantization. For a lattice this amounts to a minimization of the quantity G(L) = 1 n 1 det(L)1/2+1/n ż DV(L) }x}dx. We contribute to the problem of finding and identifying the best lattices for this measure of quality, by utilizing a reduction theory of Voronoi. This will allow us to reduce the general problem of evaluating the integral above for arbitrary lattices, to a finite number of easier problems. To be precise we will be able to express this integral as the quotient of a polynomial by a power of det(L) in Corollary 6.3.9. This expression also will show that the problem of finding the best lattice in a fixed dimension can be translated into a finite number of polynomial optimization problems in Theorem 6.3.7. But beware, their number explodes from dimension 6 onwards. The methods utilized do not depend on the underlying point set to be a lattice. More generally they work for periodic sets and, in particular, for periodic sets for which all Dirichlet-Voronoi cells are congruent to one another. This latter property is commonly asked for in quantization, following a conjecture of Gersho [Ger79]. We thus chose to derive all results directly in the case of monohedral periodic sets, including lattices as a special case. The approach to obtain an explicit formula for the quantizer constant in this way is not new, it was first used in [BS83] to prove that the lattice A#3 – D#3 is a global minimum for the lattice quantizer problem in dimension 3, and that it is in fact the only local minimum in this dimension. We provide a systematic treatment of this approach and find that both A#4 and D # 4 are local minima in dimension 4 in Theorems 6.4.3 and 6.4.4. STRUCTURE OF THE THESIS This thesis is divided into two parts, which are almost independent of each other, only Part II uses some of the notation provided in Chapter 1 of Part I to reduce redundancy. This is done on purpose to make the main results of each part easier to access if one is only interested in this particular problem. We described above how these parts are connected and that, in fact, they started out from a single question: what can mathematics contribute to the vector quantizer problem? Part I starts off with an introductory chapter on quadratic forms and lattices (cf. Chapter 1). We then present some results on maximal totally isotropic submodules of regular quadratic modules over finite rings (cf. Chapter 2). The first part is concluded by an arithmetic approach to the enumeration of (genus-)similar sublattices of integral lattices (cf. Chapter 3). Part II starts off with an introductory chapter on point sets and tilings in Euclidean space (cf. Chapter 4). Here we continue with a short introduction to, or survey of, vector quantization (cf. Chapter 5). The second part is concluded by the application of Voronoi ’s second reduction theory to the quantizer problem for monohedral periodic sets and local optimality of lattices in dimension 4 (cf. Chapter 6). vii INTRODUCTION MAIN RESULTS We present a list of what we deem to be our main results, by parts. They are presented in their order of appearance. Part I: • Theorem 2.4.4, achieving the full classification of maximal totally isotropic submodules of regular quadratic modules over finite local rings. • 2.6, presenting the explicit numbers of maximal totally isotropic submodules of regular quadratic modules over finite local principal ideal rings, in low dimensional cases. Those are of importance for the application to the enumeration of similar sublattices. • Theorem 3.3.13, the most general result we derive on the validity of the arithmetic approach to enumerate (genus-)similar sublattices of integral maximal lattices. Part II: • Theorem 6.3.7, showing that the quantizer problem for suitable periodic sets is equivalent to a finite number of polynomial optimization problems. • Corollary 6.3.9, providing an explicit expression for the quantizer constant for lattices. • Theorems 6.4.3 and 6.4.4, proving local optimality of the root lattices D#4 and A#4 for the lattice quantizer problem in dimension 4. USE OF COMPUTER ALGEBRA SYSTEMS We used the computer algebra systems MAGMA [BCP97] (V2.22-7) and MAPLE3 (2015.0) [Map] to obtain the computational results of this thesis. Wherever results depend on the usage of said systems, an indication is given. 3Maple is a trademark of Waterloo Maple Inc. viii List of notations We provide a list of symbolic notations used throughout this thesis. The symbolic notation is grouped according to the chapter it is introduced in and sorted in order of appearance. Chapter 1 bq 1.1.1 the associated bilinear form of the quadratic form q. [¨], x¨y matrix notations for free quadratic modules. O(E,b), O(E,q) 1.1.2 isometry group of the (E,b) (E,q). – isometry of quadratic and bilinear modules. GB(E,b), GB(E,q) 1.1.3 gram matrix of (E,b), (E,q). d(E,b), d(E,q) determinant of (E,b), (E,q). GB(E,b), GB(E,q) 1.1.3 gram matrix of (E,b), (E,q). K 1.1.4 as operator: orthogonal sum; as exponent: orthogonal module. E˚ 1.1.5 the dual module of E, E˚ := HomR(E,R). bˆ b on E: bˆ : EÑ E˚; y ÞÑ bˆ(y). bˆ(y) b on E: bˆ(y) : EÑ R; x ÞÑ b(x,y). H 1.1.6 the hyperbolic plane. Qp, Zp 1.2.1 the field (ring) of p-adic rational numbers (integers). Q∞, Z∞ the Archimedean completion of Q, Z. Sp 1.2.2 the Hasse-symbol. sL, nL, vL 1.3.2 the scale, norm, volume ideal of the lattice L. αL the lattice L with its quadratic (bilinear) form scaled by α. L# the dual lattice of L, L# := tx P KL | b(x,L) Ă ou. Lp,# the p-partial dual lattice of L, Lp,# := 1 p X L#. Lp the localization of L at p. gen(L) the genus of L. gL 1.3.4 the norm group of L. ix LIST OF NOTATIONS Chapter 2 We abbreviate: mtis. =ˆ maximal totally isotropic submodule(s) spec(R) 2.2.1 the spectrum of a ring. m a maximal ideal of a local ring. Rm localization of a ring at m. , [k] 2.2.4 reduction modulo m, mk. W non-hyperbolic module of rank 1, 2. νm(v) 2.3.1 m-order of v, νm(v) := max k P N0 mr´kv = 0 ( . V(k), V pr (k) 2.3.1 set of elements of m-order ě, = k. S(V), S˚(V) 2.3.2 the sets of isotropic, primitive isotropic elements of V . s(V), s˚(V) 2.3.2 cardinalities of the sets S(V), S˚(V) s˚(V)k number of isotropic elements with m-order k. νM 2.4.2 M submodule, νm(m) m PM : q(m) = 0 ( . He,f(k), He,f(k) mtis of hyperbolic planes. t a type. m(V), m(V , t) number of mtis, such of type t. s˚(t)k number of isotropic elements with m-order k in a mtis of type t. Chapter 3 Σ(L) 3.1 the set of similarities of L. SSL(L) the set of similar sublattices of L. sL, sslL counting functions for similarities, similar sublattices. Σg(L) the set of genus similarities of L. SSLg(L) the set of genus-similar sublattices of L. sgL, ssl g L counting functions for genus similarities, genus-similar sublattices. Chapter 4 We abbreviate: D-V =ˆ Dirichlet-Voronoi FB(L) 4.1 the fundamental parallelotope of L with respect to the basis B. Λt standard periodic set Λt := Ťn i=1 ti + Zn. volb 4.1.1 volume with respect to an inner product b. vol(L,b) vol(L,b) := volb(FB(L)) = a det(L,b). t 4.1.2 a collection of translation vectors. GLtn(Z) tU P GLn(Z) | UΛt = Λtu. Sną0 4.1.3 the cone of positive-definite symmetric quadratic matrices. Sn,mą0 4.1.5 set of positive definite periodic forms. (Q, t) a periodic form. x star(v,C) 4.2.1 the star of v in the polytopal complex C. C(P),C(BP) trivial subdivision of a polytope P, its boundary complex. vert(C) the vertices of the polytopal complex C. pv´ (C) pulling refinement of a polytopal complex. DVP,}¨}(p) 4.3.3 D-V cell of p in P with respect to } ¨ }. DVP(} ¨ }) D-V subdivision of P with respect to } ¨ }. DV(L), DV(Λ) the D-V cell of a lattice L, periodic set Λ. DV(Q) the D-V cell of the lattice (Zn,Q). DelP(} ¨ }) 4.3.4 the Delone subdivision of P with respect to } ¨ }. DV(Q,P) 4.3.13 the D-V polytope for the Delone polytope P, quadratic form Q. Chapter 5 G(L) 5.1 the normalized second moment/ quantizer constant of a lattice L. Gb(L) the normalized second moment/ quantizer constant of a lattice (L,b). Chapter 6 T 6.1.1 linear subspace of Sną0. ∆T (D) the T -secondary cone of D. M(S) 6.2.1 the matrix whose i-th row is (si ´ s0)T , si P vert(S), S a simplex. G(Q) 6.3 G(Q) := G(L,Q). G∆(Q) the quantizer polynomial (in Q) for the secondary cone ∆. c(Q) 6.4.1 conic parameter of Q. Fc a function F in conic parameters. Vn the Delone triangulation for Voronoi’s principal domain of the first type. ∆(Vn) Voronoi’s principal domain of the first type. xi PART I SIMILAR SUBLATTICES AND FINITE QUADRATIC MODULES 1 CHAPTER 1 Quadratic modules and lattices In this first introductory chapter we collect basic algebraic and arithmetic notions and results of the theory of quadratic forms. We first discuss the abstract theory of quadratic forms over rings and fields in 1.1. This includes the notions of regularity, duality and isometry. Basic facts about the localizations of Q and Z in 1.2 are followed by a discussion of lattices over Z and its localizations in 1.3. In particular, we provide an overview of the arithmetic theory of integral lattices and their localizations, including the notion of a genus of a lattice and the classification of unimodular and maximal Zp-lattices. 1.1 QUADRATIC FORMS OVER RINGS AND FIELDS We follow the exposition in [Kne02]. Let R be a commutative ring with 1 and let E be an R-module. Furthermore let S be a ring extension of R, i.e., there exists a monomorphism of rings ι : R ãÑ S. 1.1.1 Quadratic and bilinear forms An S-valued quadratic form on E is a map q : EÑ Swhich satisfies q(λx) = λ2q(x) for all λ P R, x P E, q(x+ y) = q(x) + q(y) + bq(x,y), where bq is a symmetric bilinear form, the associated bilinear form to q. The pair (E,q) is then called a quadratic module. If R is a field, E is a vector space and we use the term quadratic space. Accordingly we speak of a bilinear module (E,b), whenever b is a symmetric bilinear form on E. Directly from the definition we obtain that 2q(x) = bq(x, x), thus if 2 is not a zero-divisor in R, a quadratic form is uniquely determined by the associated bilinear form bq, if 2 P Rˆ, we obtain q(x) = 1 2bq(x, x). In this case only little distinction is necessary between (E,q) and (E,bq). 3 1. QUADRATIC MODULES AND LATTICES If E is a finitely generated projective1 R module of rank n, we can find a not necessarily symmetric bilinear form a : Eˆ EÑ R for which q(x) = a(x, x). We can do so by choice of a basis (x1, . . . , xn) for a free module G = E‘ E 1 enveloping E, using q( nÿ i=1 λixi) = nÿ i=1 q(xi)λ 2 i + ÿ 1ďiăjďn bq(xi, xj)λiλj, and setting a(xi, xi) := q(xi), a(xi, xj) := bq(xi, xj) for i ă j and a(xi, xj) := 0 for i ą j. We silently used the extension of q to G where q(x) = 0 for all x P E 1. In any case, a bilinear form a will always lead to a quadratic form q(x) := a(x, x), with associated bilinear form bq(x,y) = a(x,y) + a(y, x). This also shows that free quadratic modules are in correspondence to homogeneous polynomials with coefficients in R of degree 2 in n indeterminates, which is the more classical definition of a quadratic form. To describe a free quadratic module of rank nwe may write a11 a12 . . . a1n a22 . . . ann  =  q(x1) bq(x1, x2) . . . bq(x1, xn) q(x2) . . . q(xn)  , where all entries below the diagonal are zero. If the matrix is diagonal we write [q(x1), . . . ,q(xn)]. If 2 is not a zero divisor we writeC b11 . . . b1n ... ... bn1 . . . bnn G = C bq(x1, x1) . . . bq(x1, xn) ... ... bq(xn, x1) . . . bq(xn, xn) G and accordingly xbq(x1, x1), . . . ,bq(xn, xn)y. 1.1.2 Isometries An isometry of quadratic modules (E,q), (E 1,q 1) is an injective module-homomorphism φ : E Ñ E 1 such that q 1(φ(x)) = q(x), @x P E. If there is a bijective isometry, we say that (E,q) and (E 1,q 1) are isometric and write (E,q) – (E 1,q 1). The collection of all isometries (E,q)Ñ (E,q) is the isometry group of (E,q), we write O(E,q). Similarly an isometry of bilinear modules (E,b), (E 1,b 1) is an injective module-homomorphism φ : EÑ E 1 such that b 1(φ(x),φ(y)) = b(x,y), @x,y P E. If there is a bijective isometry, we say that (E,b) and (E 1,b 1) are isometric and write (E,b) – (E 1,b 1). 1A module is (finitely generated) projective if and only if it is a direct summand of a free module (of finite rank), cf. Propo- sition 3.10 and its Corollary in [Jac09]. 4 1.1. QUADRATIC FORMS OVER RINGS AND FIELDS The collection of all isometries (E,b)Ñ (E,b) is the isometry group of (E,b), we write O(E,b). It is readily checked that each isometry of quadratic modules is also an isometry for the associated bilinear forms. 1.1.3 Gram matrices and determinants of free modules If (E,b) is a free bilinear module of rank n and B = (x1, . . . , xn) is a basis, the matrix GB(E,b) := (b(xi, xj))i,j=1,...,n is the Gram matrix of (E,b) with respect to the basis B. If B 1 is another basis there is an element T P GLn(R) such that GB1(E,b) = TTGB(E,b)T . To any Gram matrix we associate its determinant in the usual way and observe that det(GB1(E,b)) = det(T)2 det(GB(E,b)). Therefore we define the determinant of a free bilinear module as the class of the determinant of any Gram matrix in R/Rˆ2. We denote this square class by d(E,b), or in short dE. We associate a determinant to any free quadratic module (E,q) by d(E,q) := d(E,bq). 1.1.4 Orthogonality Elements x,y P E are orthogonal with respect to some symmetric bilinear form b : E ˆ E Ñ R if b(x,y) = 0. For any F Ă E we have an orthogonal submodule FK := ty P E | b(F,y) = 0u. We say that E is the (internal) orthogonal sum of submodules E1, . . . ,Em if E = Àm i=1 Ei and b(Ei,Ej) = 0 for all i ‰ j, and write E = E1 K . . . K Em. The (external) orthogonal sum of bilinear modules (E,b), (E 1,b 1) is their exterior direct sum, together with the bilinear form b K b 1 : E‘E 1ˆE‘E 1 Ñ R; (x+ x 1,y+ y 1) ÞÑ b(x,y) + b 1(x 1,y 1). It is denoted by (E K E 1,b K b 1). For a quadratic module (E,q) we use the above terms if they hold for the associated bilinear form bq. 1.1.5 Duality and regularity We denote the dual module of any R-module E by E˚ := HomR(E,R). Assume that E is equipped with a bilinear form b. We define a module homomorphism bˆ : E Ñ E˚, where the value of bˆ on y P E is given by the linear form bˆ(y) : EÑ R; x ÞÑ b(x,y). To any submodule F Ă Ewe write bˆF for the map bˆ followed by the restriction map from E˚ to F˚, that is, the value of bˆ on y P E is given by the linear form bˆF(y) : FÑ R; x ÞÑ b(x,y). It is clear that ker(bˆF) = FK. For a submodule F Ă E we therefore have E = F K FK if and only if bˆF induces a bijection of F on bˆF(E), that is if bˆF(E) = bˆF(F) and FX FK = t0u. We write bˆ = bˆE. We say that a bilinear form b on E is non-degenerate if bˆ is injective, and regular if bˆ is bijective and E is finitely generated projective. A submodule F of a module E is primitive if F is a direct summand of E. If b is a bilinear form on E, F is b-primitive if F is finitely generated projective and if pbF(E) = F˚. In this situation, a regular submodule is b-primitive and a b-primitive submodule is primitive. 5 1. QUADRATIC MODULES AND LATTICES For a quadratic module (E,q) we use the above terms if they hold for the associated bilinear form bq. 1.1.6 Isotropy and hyperbolic modules Let (E,q) be a quadratic module. x P E is called isotropic if q(x) = 0, otherwise it is called anisotropic. A submodule F Ă E is called isotropic if it contains an isotropic x ‰ 0, it is called anisotropic if it does not contain non-zero isotropic elements. It is called totally isotropic if every element is isotropic. For a free R-module E setH(E) := E‘E˚ and equip it with the quadratic form q(x,y) :=ă x,y ą= y(x). This defines a free regular quadratic module. We say that a free quadratic module (E,q) is hyperbolic if there exists some free R-module F such that E – H(F). If (E,q) – H(R) we say that (E,q) is a hyperbolic plane. A hyperbolic plane can be written as Re + Rf, with q(e) = q(f) = 0 and bq(e, f) = 1, we then refer to such a basis (e, f) as hyperbolic pair. In the above matrix notation this is expressed by [ 0 1 0 0 ] and B 0 1 1 0 F , whenever applicable. 1.2 LOCAL NOTIONS References for the subsequent discussion should be found in almost all introductory texts on algebraic number theory and the arithmetic of quadratic forms, for example Kapitel 2 in [Neu06] and Section 15 in [Kne02]. 1.2.1 Localizations of Q and Z Every valuation on Q is equivalent to either a p-adic valuation, where p runs through the primes of Z, or to the usual absolute value (cf. 12.1 in [O’M73]). It is common to say that a p-adic valuation is introduced by the finite (or non-Archimedean, or prime) spot p (or rather p = (p) = pZ) and the absolute value is induced by the infinite (or Archimedean) spot ∞. We will denote spots on Z by p, including the case p =∞. If p is a prime of Z, we denote by Qp the completion of Q in the topology induced by the p-adic valuation, and refer to it as the localization of Q at p, it therefore is a complete local field. Similarly we write Zp for the completion of Z under the above topology and refer to this as the localization of Z at p. In this situation Qp is the field of fractions of Zp and Zp is the valuation ring and topological closure of Z inQp, both with respect to the p-adic valuation. We refer to elements of Zp as the integers of Qp. For a fractional ideal a of Z we can define ap as the topological closure of a in Zp (cf. 81 : 13 in [O’M73]). Note that Zp is a discrete valuation ring and therefore a principal ideal domain. If p is the Archimedean spot∞ the situation becomes even simpler, we set Q∞ = R, the completion of Q with respect to the usual absolute value. We then write Z∞ = Q∞ = R. 6 1.3. LATTICES OVER Z AND ITS LOCALIZATIONS 1.2.2 Local invariants for quadratic spaces We define the Hilbert-symbol for Q and primes p P Z to be (a,b)p := # +1, Dx,y P Qp : ax2 + by2 = 1, ´1, else. Clearly the Hilbert-symbol is invariant under distinct representatives of a square class in each argu- ment, thus we will allow for the arguments to be square classes of field elements. If V –ă α1 ąK . . . Kă αn ą is a non-degenerate quadratic space of dimension n overQp we also have the Hasse-symbol SpV = ź 1ďiďn (αi, ź 1ďjďi αj)p. This definition does not depend on the chosen orthogonal splitting of V . Furthermore if p is a prime spot and V 1 is another Qp space of dimension n, then V – V 1 if and only if SpV = SpV 1 (cf. §63B. in [O’M73] ). The only invariant we need for quadratic spaces over Q∞ is the usual notion of the signature over the real numbers. 1.3 LATTICES OVER Z AND ITS LOCALIZATIONS We now gather aspects of the theory of lattices over the ring Z or one of its completions and clarify the notation used. A general reference for this is [O’M73] and we follow closely the notation of this book and only indicate certain spots of interest more clearly with a direct reference to this book. Let K be one of the fields R,Q,Qp,Q∞, where p is a prime. Let o = Z if K P tR,Qu and let o be the localization of Z in K if K = Qp for a prime p or K = Q∞ = R. 1.3.1 Lattices An o-lattice in a K vector space V is a finitely generated o-submodule of V , it is on V if KL = V . For any lattice in V we can find a basis B, by which we understand a vector space basis B = (v1, . . . , vm) of KL Ă V , such that L = ov1 + . . . + ovm. The number m = dim(KL) = rk(L) is the rank of L as o-module and is referred to as the rank of the lattice L. All of the above lattices are free modules, because we assume o to be a principal ideal ring. A subset L 1 Ă L that is itself a lattice, is called a sublattice of L. 1.3.2 Lattices on bilinear and quadratic spaces Let (V ,b) is a bilinear space with associated quadratic form q defined by q(x) = b(x, x)2 for x P V . Let L be a lattice on V3. If B = (v1, . . . , vn) is a lattice basis, the Gram matrix of L with respect to B is the 2That is, b = 12bq. 3It is not hard to see that we in fact could start with some finitely generated torsion-free quadratic or bilinear o-module L and embed it into the K vector space V := Kbo L such that L is a lattice on V . This shows that the actual object of interest is given by the pair (L,b), resp. (L,q). 7 1. QUADRATIC MODULES AND LATTICES Gram matrix of the quadratic o-module (L,q) GB(L,b) := GB(V ,b) = (b(vi, vj))i,j=1,...,n. The volume ideal of L is the ideal vL = det(GB(L,b))o, this definition does not depend on the chosen basis: the determinants of different Gram matrices differ by the square of a unit. The determinant of L is the determinant of the quadratic o-module (L,q), thus vL is the ideal generated by any representative of the determinant of L. If o = Z, 1 is the only square of a unit in Z. We therefore interpret det(L,q) as an integer and obtain vL = (det(L)). For a lattice L a Gram matrix of V with respect to any lattice basis B is also called a Gram matrix of L, we write GB(L,b). If α P K and L is in (V ,b), we write αL for the lattice L in (V ,αb), which we abbreviate to αV if the bilinear form is clear from context. We say that we scale L (respectively V , or b) by α in that case. To L we associate the scale ideal sL, the o-ideal generated by the set b(L,L) = tb(x,y) | x,y P Lu, and the norm ideal nL, the o-ideal generated by the set q(L) = tb(x, x) | x P Lu. Then (cf. §82E. in [O’M73]) 2sL Ă nL Ă sL, and vL Ă (sL)n. Furthermore for α P K we immediately obtain sαL = αsL, nαL = αnL, vαL = αnvL. Let L be a Z-lattice over K P tQ,Ru and let L 1 be a sublattice of L. Then det(L 1) = [L : L 1]2 det(L), this is sometimes called the determinant-index formula (cf. 81 : 11 and 82 : 11 in [O’M73]). A non-zero lattice on a non-degenerate quadratic space (V ,b) is called a-maximal, for some fractional ideal a, if nL Ă a and for every lattice L 1 on V , for which L Ă L 1 and nL 1 Ă a are satisfied, the equality L = L 1 holds. For α P K we immediately obtain that if L is a-maximal on V , then αL is αa-maximal on αV . We can adapt the notions of isometry and isometry groups to lattices in a literal manner from 1.1.2. This induces an equivalence relation on the set of lattices on V and the equivalence class of a lattice L is called isometry class of L, denoted by cls(L). The isometry group of a lattice is often called its orthogonal group if o = Z and K P tQ,Ru. The automorphism group of L is Aut(L) = tφ P GL(V) | φ(L) = Lu, thus O(L,b) = Aut(L)XO(V ,b). If the bilinear form or quadratic form is clear from context we also abbreviate O(L) for O(L,b) or (O(L,q)). For α P K we immediately obtain O(L) = O(αL), and cls(L) = cls(αL). The dual lattice of L in (V ,b) is given by L# := tx P KL | b(x,L) Ă ou. If B = (v1, . . . , vm) is a basis for L in which L = ov1 + . . . + ovm, 8 1.3. LATTICES OVER Z AND ITS LOCALIZATIONS then L = ov#1 + . . . + ov # m, where B# = (v#1, . . . , v # m) is the dual basis of B with respect to b, that is b(vi, v#j) = δij. Let L be an integral Z-lattice over Q or R. We associate to any integer p | det(L) the p-partial dual Lp,# := 1 p LX L# and for any co-prime integers p,q such that det(L) = pq we immediately derive L# = Lp,# X Lq,#. (1.1) A lattice is called a-modular for some fractional ideal a if sL = a and vL = an. For α P K we say that L is α-modular if L is (α) = αo-modular. L is unimodular if it is o-modular. If (L,b) is non-degenerate then L is unimodular if and only if L = L#. 1.3.3 Localizations and the genus of an integral lattice Let p be a prime of Z. We can localize a vector space V overQ and a Z-lattice L in V . We set Vp := QpV and Lp = ZpL for the localization. In addition, Q∞ = R and Z∞ = R. Then Lp is a lattice on Vp, and in fact Lp is the Zp-submodule generated by L in Vp. If B is a basis of L, then Lp = Zpv1 + . . . + Zpvm. In particular s(Lp) = (sL)p, n(Lp) = (nL)p, v(Lp) = (vL)p, and L is a-maximal if and only if Lp is ap-maximal (cf. §82K. [O’M73]). It is the well-known Hasse-Minkowski Theorem (cf. 66 : 4 in [O’M73]) that relates non-degenerate quadratic spaces and their localizations: non-degenerate quadratic Q-spaces V ,V 1 are isometric if and only if Vp – V 1p for all primes p of Z and V∞ – V 1∞. Therefore, two non-degenerate Q-spaces V ,V 1 are isometric if and only if Vp – V 1p for all prime numbers and if the signature ofRV andRV 1 are identical. Now for lattices on a non-degenerate quadratic spaceV the Hasse-Minkowski Theorem fails in general: there might be non-isometric lattices L,L 1 such that Lp – L 1p for all spots p (including primes and∞) of Z. Thus the set gen(L) := tL 1 lattice on V | L 1p – Lp for all spots p of Zu will in general be a disjoint union of several distinct isometry classes of lattices on V . We call this set the genus of L. The number of distinct isometry classes in gen(L) is the class number of L and we say that L is unigeneric if this number is equal to 1. 1.3.4 Unimodular lattices over local fields Let L be a Zp-lattice over Qp, where p is a prime of Z. In this situation, for any dimension n, the classification of unimodular lattices divides into two cases. We have to handle the case that p | 2, the so called dyadic case and the case that 2 P Zp˚, the so called non-dyadic case, separately In the non-dyadic case there are two classes of non-isometric non-degenerate quadratic spaces overQp on which a unimodular lattice can exist, and on each of those there exists exactly one isometry class of 9 1. QUADRATIC MODULES AND LATTICES unimodular lattices. In particular, this implies that if L,L 1 are on V and both are unimodular at p, then Lp – L 1p (cf. §92. in [O’M73]). In the dyadic case this becomes more difficult. We need an additional invariant, the norm group of a lattice gL := q(L) + 2sL. This group in fact generates the norm ideal, but might provide more information as 2sL Ă gL Ă nL. And in fact: unimodular lattices L,L 1 on the same space over Qp are isometric if and only if gL = gL 1 (cf. Theorem 93 : 16 in [O’M73]). For unimodular lattices over Z2 this implies that g(L) = Z2 or g(L) = 2Z2, the odd and even case. 1.3.5 Maximal lattices over local fields Another well-behaved class of lattices over the fields Qp is that of maximal lattices. If V is a regular quadratic space overQp and if a is a fractional ideal of its ring of integers, then L – L 1 for all a-maximal lattices L,L 1 on V (cf. 91 : 2 in [O’M73]). 10 CHAPTER 2 Maximal totally isotropic submodules of regular quadratic modules over finite rings This chapter contains some of the main results of this thesis. We provide a local-global principle for quadratic forms over finite rings, classify maximal totally isotropic submodules of regular quadratic modules over finite rings, and count their number over finite principal ideal rings. We start by recalling some basic results for quadratic forms over special rings. This includes splitting off hyperbolic planes, Witt’s Theorem over local rings, and basic facts regarding quadratic forms over finite fields. This is followed by a discussion of quadratic forms over finite rings, where we derive a local-global principle for finite quadratic forms in Theorem 2.2.2. Using a variant of Hensel’s Lemma we relate regular quadratic modules over finite local rings to regular quadratic spaces over their residue fields. As a consequence, we can divide regular quadratic modules over finite local rings into 3 cases (cf. Proposition 2.2.5). For each case we can derive a formula for the order of the orthogonal group (cf. 2.2.8). Together with a construction we refer to as slicing (cf. 2.3.1), this enables us to give explicit formulae for the number of isotropic elements of given order (cf. Propositions 2.3.8 and 2.3.9). We then focus on the classification of maximal totally isotropic submodules of regular quadratic mod- ules over finite rings (cf. 2.4). This classification is one of the main results of this thesis and achieved in Theorem 2.4.4. Its proof is subdivided into a number of Lemmata and presented in 2.4.3. The classifi- cation is based on the new notion of type of a maximal totally isotropic submodule, explicit examples of such submodules, for every type, are collected in Example 2.4.3. Combining the results of 2.3 and 2.4, we provide a way to count the number of maximal totally isotropic submodules of regular quadratic modules over finite principal ideal rings. The main case is that the ring is also local, then Proposition 2.5.5 gives the number of maximal totally isotropic sub- modules of a fixed type. The general case can be reduced to the local case by the local-global principle established earlier (cf. Proposition 2.5.1). 11 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS We conclude this chapter with some explicit numbers of maximal totally isotropic submodules of reg- ular quadratic modules of even rank over finite local principal ideal rings (cf. 2.6). In particular, these results apply to modules over Z/prZ (and by multiplicativity to modules over Z/cZ). By Theorem 3.3.12) we find that (2.9) provides the number of similar sublattices of E8 of norm pr. These were not known before, this application to the counting of similar sublattices was the main motivation for the subsequent research. 2.1 GENERALITIES 2.1.1 Quadratic modules split off hyperbolic planes For the sake of readability we collect some basic results for quadratic modules over commutative rings with 1. PROPOSITION 2.1.1 (CF. (1.6) IN [KNE02]). Let (V ,q) be a quadratic module. If U Ă V is a regular sub- module then U splits V : V = U K UK. As in the case of fields, a free regular totally isotropic submodule of a quadratic module can be em- bedded in a hyperbolic submodule. A bit more is actually true. PROPOSITION 2.1.2 (CF. (2.22) IN [KNE02]). Let U be a bq-primitive free totally isotropic submodule of a quadratic module (V ,q). Then there exists a hyperbolic submoduleW Ă V such that U ĂW. In the case of primitive isotropic elements of regular modules, we obtain the well known process of splitting off an hyperbolic plane. COROLLARY 2.1.3. Let (V ,q) be a quadratic module. Let v P V be bq-primitive isotropic. Then there exists a hyperbolic plane H and a submoduleW Ă V such that V = H KW 2.1.2 Witt’s Theorem for quadratic forms over local rings Let R be a local ring with maximal ideal m. We start by recalling in which way a quadratic module over a local ring can be described as an orthogonal sum of submodules. PROPOSITION 2.1.4 ((CF. (4.1) IN [KNE02]). Let V be an finitely generated R-module with symmetric bilin- ear form b. Then V can be decomposed orthogonally into regular submodules of rank 1 or 2, and a module W such that b(W,W) Ă m. V is regular if and only ifW = t0u. There is a version of Witt’s theorem which holds over local rings: THEOREM 2.1.5 (CF. (4.3) IN [KNE02]). Let (V ,q) be a quadratic R-module with associated bilinear form b = bq, X, Y,Z submodules, where X, Y are free and of finite rank and let bˆX(Z) = X ˚, bˆY(Z) = Y˚ be satisfied. Let φ : XÑ Y be an isomorphism satisfying φ(x) ”Z x for all x P X. Then there is an extension of φ to an automorphism of V , satisfying φ(v) ”Z v for all v P V , while fixing every z P ZK. 12 2.1. GENERALITIES This implies the validity of Witt’s cancellation Theorem for local rings. COROLLARY 2.1.6. Let (V ,q) be a quadratic module over a local ring R. Let φ 1 : X Ñ Y be an isometry of bq-primitive submodules of V . Then there is an isometry φ : V Ñ V such that φ|X = φ 1. COROLLARY 2.1.7. Let (X1,q1), (X2,q2), (Y1,q1), (Y2,q2) be quadratic modules over a local ring R, where X1,X2 are regular. Then X1 K Y1 – X2 K Y2 and X1 – X2 =ñ Y1 – Y2. COROLLARY 2.1.8. Let (V ,q) be a quadratic R-module. Then rk(M1) = rk(M2) for any bq-primitive maxi- mal totally isotropic submodulesM1,M2 of V . We will write ind((V ,q)) := rk(M) for any bq-primitive maximal totally isotropic submodule of a quadratic module (V ,q) and refer to this as the Witt index of (V ,q), which is well-defined by the above result. 2.1.3 Facts on quadratic forms over finite fields Let (V ,q) be a regular quadratic space with ind(V) = m over Fq. The following facts can be found in §12 Klassifikation and §13 Anzahlbestimmungen in [Kne02]. FACT 2.1.9. Let V be a regular quadratic Fq-space with ind(V) = m. Then V decomposes as V – H K . . . K H KW, withW anisotropic, where we distinguish 3 cases: I) dim(V) = 2m, dim(W) = 0 and V is hyperbolic. II.1) dim(V) = 2m+ 1, dim(W) = 1 and V is non-hyperbolic of odd rank, II.2) dim(V) = 2m+ 2, dim(W) = 2 and V is non-hyperbolic of even rank. In cases I and II.2 the decomposition determines the isometry class of V . In case II.1 and char(Fq) ‰ 2, there are 2 distinct isometry classes. In all these cases the spaces are regular. If char(Fq) = 2 there exist no regular quadratic spaces of odd dimension. In the case of char(Fq) ‰ 2 this can be reformulated in terms of orthogonal bases. FACT 2.1.10. Let (V ,q) be a regular quadratic Fq-space where char(Fq) ‰ 2. Then V –ă 1, . . . , 1, det(q) ą, and there are two distinct isometry classes of given dimension. FACT 2.1.11. Let V be a regular quadratic Fq-space with ind(V) = m. The order of the corresponding orthog- onal groups are I) |O(V)| = 2 ¨ qn(n´1)/2 ¨ (1´ q´m) ¨ ź 2iăn (1´ q´2i), II.1) |O(V)| = c ¨ qn(n´1)/2 ¨ ź 2iăn (1´ q´2i), II.2) |O(V)| = 2 ¨ qn(n´1)/2 ¨ (1 + q´m´1) ¨ ź 2iăn (1´ q´2i), where c = 1 if char(Fq) = 2 and c = 2 else. 13 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS FACT 2.1.12. Let V be a regular quadratic Fq-space with ind(V) = m. The number of primitive isotropic vectors is given by I) s˚(V) = (qm ´ 1)(qm´1 + 1), II.1) s˚(V) = q2m ´ 1, II.2) s˚(V) = (qm+1 + 1)(qm ´ 1). FACT 2.1.13. For regular quadratic spaces V andW, both hyperbolic or both non-hyperbolic, with dim(W) = dim(V)´ 2: |O(V)| = s˚(V) ¨ qn´2 ¨ |O(W)|. 2.2 REGULAR QUADRATIC FORMS OVER FINITE RINGS 2.2.1 Artinian rings A commutative ring R, with unity, is artinian if it satisfies the descending chain condition for ideals, that is, if for every descending chain I1 Ą I2 Ą . . . of ideals of R, there exists an r P N such that I1 Ą I2 Ą . . . Ą Ir = Ir+1 = . . . , we say that the chain becomes stable. In particular, @I ă R Dr P N : I Ą I2 Ą . . . Ą Ir = Ir+1. (2.1) Let specR denote the spectrum of R, that is, the set of all prime ideals of R. If R is artinian, every prime ideal is already maximal (cf. Proposition 8.1 in [AM94]) and specR is finite (cf. Proposition 8.3 in [AM94]). The intersection N(R) := Ş mPspecRm is the nilradical and it is nilpotent (cf. Proposition 8.4 in [AM94]). Let m P specR, and let r be minimal in (2.1). Then we get the localization Rm – R/mr of R at m (cf. proof of Theorem 8.7 in [AM94]). If R is artin local with maximal ideal m, then m = N(R) and thus nilpotent. In this case the minimal r in (2.1) is the nilpotency index of m in R. THEOREM 2.2.1 (CF. THEOREM 8.7 IN [AM94]). Let R be artinian and let r be the nilpotency index ofN(R). Then R is a finite direct product of artin local rings: R – ź mPspecR Rm – ź mPspecR R/mr. For a local artinian ring R the quotients mi/mi+1 are R/m-vector spaces. Since artinian rings are noethe- rian, the dimension of these R/m-spaces is finite. Furthermore, R is a principal ideal ring if and only if dimR/m(m/m2) ď 1. 14 2.2. REGULAR QUADRATIC FORMS OVER FINITE RINGS 2.2.2 Finite rings Every finite ring R is both artinian and noetherian and thus Theorem 2.2.1 is applicable. Let R be finite local, with maximal ideal m. Then R/m is a finite field, and thus there exists a prime p and a power q = pk such that R/m – Fq. Then there exists s P N, such that |R| = qs, in fact, |R| =śr i=1 |mi´1/mi| and as we noted each mi´1/mi is a Fq-vector space, thus all cardinalities appearing are powers of q. But then q = |R/m| = [R : m] = |R|m and thus |m| = qs´1. R is a principal ideal ring if and only if dimR/m(m/m2) ď 1. We distinguish the two possible cases. Either dimR/m(m/m2) = 0 and m = m2 = t0u, so R is a field, or |mk| = |m|k = qk for all 0 ď k ď r and, in particular, r = s. 2.2.3 A local-global principle for quadratic forms over finite rings Let R be finite, let specR = tm1, . . . ,mlu. By Theorem 2.2.1 R – lź i=1 Rmi – lź i=1 R/mrii , (2.2) where ri is minimal in (2.1) for mi. Let V be a finitely generated Rmodule. Then this implies that V – lź i=1 Vmi – lź i=1 V/mrii , where Vmi := Rmi bR V . If V carries a quadratic form q, we can localize q as well: qmi : Vmi Ñ Rmi defined by qmi(α b v) := α b q(v). The associated bilinear form qmi is then the localization of the associated bilinear form of q. Now each Rmi is a flat R-module, that is, tensoring with Rmi preserves exact sequences (cf. p. 16 and Theorem (3.11) in [Rei75]). This property is quite useful, as the next result will show. THEOREM 2.2.2 (LOCAL-GLOBAL PRINCIPLE FOR FINITE QUADRATIC FORMS). Let R be a finite ring. i. A quadratic module (V ,q) is regular over R if and only if (Vm,qm) is regular for all m P specR. ii. For R-modules (V ,q), (V 1,q 1), we have (V ,q) –R (V 1,q 1) if and only if (Vm,qm) – (V 1m,q 1m) for all m P specR. PROOF. Ad (i): We consider the dual module of V first. Let m P specR: (HomR(V ,R))mi = Rmi bR HomR(V ,R) – HomRmi (Vmi ,Rmi) – Hom R/m ki i (V/mkii ,R/m ki i ), where the first isomorphy comes from Theorem 3.18 in [Rei75] if we note that V is clearly finitely presented. Taking this isomorphy and the decomposition of R, V , and V˚ as direct products, we see that (V ,q) is regular if and only if bˆq : V Ñ HomR(V ,R) 15 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS is an isomorphism if and only if bˆqm : Vm Ñ (HomR(V ,R))m – HomRm(Vm,Rm) is an isomorphism for all m P specR if and only if (Vm,qm) is regular for all m P specR. Ad (ii): Write specR = tm1, . . . ,mlu and to each mi let ri be as in (2.2). Clearly q(v) = q 1(v 1) if and only if Φ(q(v)) = Φ(q(v 1)) under the isomorphism Φ : RÑ lź i=1 R/mrii . Let pii be the projection of the direct product onto R/mrii , and ιi its injection. Then given a map φ : V Ñ V 1 we obtain a mapΦ(φ) : śli=1 V/mrii Ñśli=1 R/mrii in the natural way. Thus, by directness of the product, we derive that φ : V Ñ V 1 is an isometry if and only if Φ(φ) is if and only if pii ˝Φ(φ) ˝ ιi : V/mrii Ñ R/mrii is. 2.2.4 Isometry classes over finite local rings We now restrict ourselves to the case that R is a finite local ring with maximal ideal m of nilpotency index r. For a quadratic R-module V we set V := V/m, q(v) := q(v) for the quadratic R/m space (V ,q) that is induced from V . More generally for 1 ď k ď r we set V [k] := V/mk, q[k](v[k]) to refer to the projection modulo the ideal mk of R, where V [1] = V and V [r] – V . We restrict our interest to finitely generated projective modules V , which then are free since R is as- sumed to be a commutative local ring (cf. [Theorem 7.7] [Jac09], also true for arbitrary projective modules over local rings, (cf. [Kap58, Theorem 2] ). Since R local, so is R/I for any ideal I ă R. In particular, m/I is the maximal ideal of R/I. It is easily seen that if V is regular over R, then V/I is regular over R/I: in fact, since V is free, regularity is equivalent to dV = εRˆ2, with ε P Rˆ. Now if v1, . . . , vn is a basis of V , then v1 + IV , . . . , vn + IV is a basis for V/I, thus dV/I = (ε+ I)(R/I)ˆ2. Thus ε P Rˆ = Rzm implies that ε+ I P (R/I)ˆ = R/Izm/I, so that the claim on regularity follows. LEMMA 2.2.3. Let (V ,q), (W,q 1) be regular quadratic R-modules of rank n. V –W ô V –W. If R is a principal ideal ring, each isometry V ÑW lifts to |m|n¨(n´1)/2 distinct isometries of V ontoW. PROOF. This is basically a finite version of Hensel’s Lemma (cf. (15.3) in [Kne02]). Let r be minimal with mr = t0u. First of all, if V – W, there is some R-linear map φ : V Ñ W, for which φ : V – W is an isometry. This can be obtained by choice of bases x1, . . . , xn of V and y1, . . . ,yn ofW for which φ(xi) = yi holds, simply set φ(xi) := yi in that case. We denote φ1 := φ for one such map and proceed iteratively. 16 2.2. REGULAR QUADRATIC FORMS OVER FINITE RINGS Given φj´1, satisfying q 1(φj´1(v)) ”mj´1 q(v) for all v P V , we construct φj satisfying q 1(φj(v)) ”mj q(v). Of course, once we reach j = r, that is after r´ 1 steps, we have constructed the desired isometry φ 1 := φr : V ÑW. Since q 1(φj´1(v)) ”mj´1 q(v), we see that the quadratic form qj(v) := q 1(φj´1(v)) ´ q(v) satisfies qj(V) Ă mj´1. Let a be a bilinear form (not necessarily symmetric) such that qj(v) = a(v, v), note that also a(V ,V) Ă mj´1 (cf. 1.1.1). Make the ansatz φj(v) = φj´1(v) +ψ(v), with some linear map ψ : V Ñ mj´1W. Then q 1(φj(v)) = q 1(φj´1(v)) + q 1(ψ(v)) + bq1(φj´1(v),ψ(v)) ”mj q 1(φj´1(v)) + bq1(φj´1(v),ψ(v)) This shows that for φj as such, by definition of qj, the congruences q 1(φj(v)) ”mj q(v) for all v P V , are equivalent to bq1(φj´1(v),ψ(v)) ”mj ´qj(v) = ´a(v, v) for all v P V . (2.3) We now claim that we can choose ψ in such a way, that more generally the following holds: bq1(φj´1(w),ψ(v)) ”mj ´a(w, v), for all w P V . To see this, fix a basis v1, . . . , vn of V . For all vi +mj we find an element zi +mj P V/mj such that bq1(φj´1(w+mj), zi +mj) + mj = ´a(w+mj, vi +mj) + mj, for all w+mj P V/mj, (2.4) simply by regularity of the reduction of bq1 modulo mj. Thus we can define a mapψ by the conditions vi ÞÑ zi. It remains to show that ψ(V) Ă mj´1W, but this follows from the fact that zi P mj´1W for i P t1, . . . ,nu: The reduction of bq1 modulo mj´1 is regular, so the congruence bq1(φj´1(w), zi) ”mj ´a(w, vi) ”mj´1 0 holds for all φj´1(w) + mj´1, where w P V . But the reduction of φj´1 modulo mj´1 is an isometry V/mj´1 Ñ W/mj´1, thus x + mj´1 ÞÑ bq1(x + mj´1, zi + mj´1) + mj´1 is the zero map, which implies zi P mj´1. It remains to count the number of extensions of every isometry over the residue field to isometries over R in the case that m ‰ t0u is a principal ideal. We have to check which zi+mj P mj´1/mj are solutions to the congruence equation (2.4). There is a total of |mj´1/mj| choices of zi for each i, thus in combination we have |mj´1/mj|n¨n possible choices for ψ modulo mj. But each ψ : V Ñ mj´1W generates one of the |mj´1/mj|n¨(n+1)/2 quadratic forms on the R/m-vector space mj´1W/mj – (mj´1/mj)n. Conversely each such quadratic form on ( mj´1/mj )n can be obtained this way, and each one of them is constructed the same number of times. This in particular implies that every quadratic form on ( mj´1/mj )n is the image of exactly |mj´1/mj|n¨(n´1)/2 = |mj´1/mj|n¨n´n¨(n+1)/2 linear maps V Ñ (mj´1/mj)n. So at each step above there are precisely |mj´1/mj|n¨(n´1)/2 linear maps V Ñ (mj´1/mj)n, such that condition (2.3) is satisfied for the quadratic form a(v, v). Completing all r´ 1 steps we arrive at rź j=2 |mj´1/mj|n¨(n´1)/2 = ( |m| |mr| )n¨(n´1)/2 = |m|n¨(n´1)/2 distinct lifts to isometries V ÑW. 17 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS REMARK 2.2.4. We have pursued the counting of lifts only in the case that m is a principal ideal. The proof is intendedly written in such a way that it can be directly generalized to the non- principal case, but more variables come in to play: as is evident from the proof, it is the dimension of the R/m vector space mj´1/mj that brings in new degrees of freedom once m is not principal. One has to evaluate this dimension at each step and the resulting quantity will not posses a closed form as simple as in the principal ideal case.  Lemma 2.2.3 provides a very important result, which allows us to explicitly describe the possible Witt- decompositions of regular quadratic modules (V ,q) over R, just as in the case of finite fields. PROPOSITION 2.2.5. Let V be a regular quadratic module over R. Then V – H K . . . K H KW, where W is the (unique up to isometry) non-hyperbolic quadratic R-module of rank 0 (case I), 1 (case II.1), or 2 (case II.2). As with finite fields, in the case of char(R) ‰ 2 we can reformulate this in terms of orthogonal bases. PROPOSITION 2.2.6. Let (V ,q) be a regular quadratic R-module where char(R) ‰ 2. Then q – x1, . . . , 1, det(q)y and there are two distinct isometry classes of given rank. PROOF. This follows from 2.2.3 and the well-known classification of regular quadratic modules over finite fields (cf. (12.5) in [Kne02] and 2.1.10). REMARK 2.2.7. The non-hyperbolic regular module W of rank 1 or 2 is not anisotropic if r ą 1, though it cannot contain primitive isotropic elements, it certainly will contain isotropic elements, namely those of the submodule mrr/2sW, which is totally isotropic (cf. Lemma 2.4.10).  By the above, it is possible to compute the orthogonal groups of regular quadratic modules over R by computation of the lifting homomorphisms used in the proof of Lemma 2.2.3, this can be of use for the construction of similar sublattices in certain cases (cf. Remark 3.3.14). In any case, we can derive the orders of the associated orthogonal groups of such modules. COROLLARY 2.2.8. Let R be a principal ideal ring. Let (V ,q) be a regular quadratic R-module of cardinality qs. Then | O(V ,q) |= |m|n(n´1)/2¨ | O(V ,q) | . That is, if V is hyperbolic | O(V ,q) |= 2 ¨ (qs)n(n´1)/2 ¨ (1´ q´n/2) ¨ ź 2iăn (1´ q´2i), if V is non-hyperbolic of odd rank | O(V ,q) |= c ¨ (qs)n(n´1)/2 ¨ ź 2iăn (1´ q´2i), where c = 1 if char(Fq) = 2 and c = 2 else, and if V is non-hyperbolic of even rank | O(V ,q) |= 2 ¨ (qs)n(n´1)/2 ¨ (1 + q´n/2) ¨ ź 2iăn (1´ q´2i). 18 2.3. COUNTING ISOTROPIC ELEMENTS IN REGULAR QUADRATIC MODULES OVER FINITE LOCAL P.I.R. 2.2.5 Isometry classes over finite rings Let R be a finite ring. From Theorem 2.2.2 we immediately derive that the isometry classes of a reg- ular quadratic module over R are in bijection with all direct products of isometry classes of regular quadratic spaces over its localizations. There are no regular quadratic modules over R if 2 | |R| and the rank of the localization at the maximal ideal containing 2 is odd (cf. 2.1.9). 2.3 COUNTING ISOTROPIC ELEMENTS IN REGULAR QUADRATIC MODULES OVER FINITE LOCAL PRINCIPAL IDEAL RINGS Throughout this section R is a finite local principal ideal ring with maximal ideal m and corresponding nilpotency index r. Note that if |R| = qs then s = r for such rings. There are a few results which hold in greater generality, these are clearly marked to hold for rings that are not necessarily principal ideal rings. A direct approach to count the isotropic elements of given m-order, comparable to 13 Anzahlbestim- mungen in [Kne02] seems to lead to a messy inductive calculation involving a range of representation numbers once R is not a field. We relate isotropic elements of fixedm-order k ă 2r to primitive isotropic elements of a related regular quadratic module over the ring R/mr´2k. Counting primitive isotropic el- ements of a regular quadratic R-module can be reduced to the counting of primitive isotropic elements over the residue field of R, combining both we obtain the number of isotropic elements of arbitrary m-order. 2.3.1 Slicing of regular quadratic modules Let (V ,q) be a regular quadratic R-module. Let v P V , we denote by νm(v) the m-order of v, that is νm(v) := max k P N0 mr´kv = 0 ( . We sort the elements of V into layers comprised of those elements with identical m-order. The union of all such layers, starting at any 0 ď k ď r, will then be a submodule. We ultimately will count the number of isotropic elements for each layer separately; to do so we proceed by giving a description of all isotropic elements with m-order equal to k below. DEFINITION 2.3.1. Let V be a regular quadratic space over R, and let 0 ď k ď r. V(k) := v P V νm(v) ě k ( , V pr (k) := v P V νm(v) = k ( . ˛ LEMMA 2.3.2. Let V be a free regular quadratic space over R, and let 0 ď k ď r. Then V(k) = m kV , V pr (k) = m kVpr. In particular for v P V , νm(v) ě k if and only if there is a v 1 such that v = µkv 1, and νm(v) = k if and only if there is a primitive v 1 such that v = µkv 1 for some µ P m. PROOF. Since V is free we can identify elements of V with the vector of their coefficients λi, with respect to some basis. Let v P V(k), that is mr´kv = 0 and it follows that for each coefficient λi we have mr´kvi = 0. But this implies that νm(λi) ě k, and λi P mk, therefore V(k) Ă mkV , the other inclusion is obvious. 19 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS Now assume νµ(v) = k. Then νµ(v 1) = νµ(v)´ k = k´ k = 0 shows that v 1 is primitive. We want to use the fact that every element of V(k) therefore “essentially is divisible by mk”. Since V(k) is annihilated by mr´k, we see that V(k) has a structure as R/mr´k-module. If 2k ă r, we make V(k) a quadratic R/mr´k-module by defining a quadratic form q(k) on V(k), that is “essentially the form q divided by m2k”. Let v1, . . . , vn be a basis of V . Without loss of generality we assume that the basis is chosen such that with respect to this basis i. for 2 - char(R) we have q = x1, . . . , 1, δywhere δ = det(q) (cf. Proposition 2.2.6); ii. for 2 | char(R) we have a decomposition V = [ 0 1 0 0 ]ind(V) KW, where W = $’’&’’% t0u if n = 2 ind(V),[ 0 1 0 0 ] if n = 2 ind(V) + 2, If 2k ă r, we fix the basis of the R/mr´k-module V(k) that is obtained by the images of v1, . . . , vn under the structure transport discussed above. By the values on this basis we define a form q(k) on V(k) as follows: i. if p ‰ 2 we define (V(k),q(k)) to be the quadratic R/mr´k-module V(k) = x1, . . . , 1, δ[r´k]y, where δ [r´k] is the canonical image of det(q) embedded in R/mr´k and the diagonal form is written with respect to the above fixed basis of V(k); ii. if p = 2 we define (V(k),q(k)) to be the quadratic R/mr´k-module V(k) = [ 0 1 0 0 ]ind(V) KW(k), where W(k) = $’’&’’% t0u if n = 2 ind(V),[ 0 1 0 0 ] if n = 2 ind(V) + 2, with respect to the above fixed basis of V(k). LEMMA 2.3.3. Let V be a regular quadratic R-module and let 2k ă r. With the above definition of q(k) we obtain a regular quadratic R/mr´k-module (V(k),q(k)). In particular, (V(k),q(k)) is hyperbolic (resp. non- hyperbolic) if and only if (V ,q) is hyperbolic (resp. non-hyperbolic). 20 2.3. COUNTING ISOTROPIC ELEMENTS IN REGULAR QUADRATIC MODULES OVER FINITE LOCAL P.I.R. By reduction of (V(k),q(k))modulomr´2k/mr´k we obtain a regular quadratic R/mr´2k module, which characterizes the isotropic elements of m-order k. LEMMA 2.3.4. Let (V ,q) be a regular quadratic R-module and let 2k ă r. The following are regular quadratic modules of the same rank. One of the three is hyperbolic if and only if all of them are. i. (V ,q) as R-module, ii. (V(k),q(k)) as R/mr´k-module, iii. ( V(k) [r´2k] ,q(k)[r´2k] ) as R/mr´2k-module. An element v P V(k) is q-isotropic if and only if νm(q(k)(v)) ě r ´ 2k if and only if v[r´2k] is q(k) isotropic. If 2k ě r, then (V(k),q(k)) is a totally isotropic quadratic R/mr´k-module and therefore every element is isotropic. Combining this we obtain the desired description of isotropic elements of m-order k. LEMMA 2.3.5. In the above notation: For v P V with 0 ď νm(v) = k ď t r2 u q(v) = 0 ô q(k)[r´2k](v[r´2k]) = 0. In addition, every v P V with νm(v) ě r r2 s is isotropic. 2.3.2 Counting elements DEFINITION 2.3.6. Let V be a regular quadratic module over R. S(V) := v P Vzt0u q(v) = 0 ( s(V) := |S(V)| S˚(V) := v P Vzt0u q(v) = 0, v primitive ( s˚(V) := |S˚(V)| ˛ We compute s˚(V) and s(V). The next Lemma shows that s˚(V) = |O(V)|/| stabO(V) |. Then it will suffice to compute | stabO(V)(v)| for an arbitrary primitive isotropic v, since we can get the value of |O(V)| from Corollary 2.2.8. LEMMA 2.3.7. Let (V ,q) be a regular quadratic R module, R not necessarily a principal ideal ring. O(V) acts transitively on S˚(V). PROOF. We fix some v0 P S˚(V), thus v0 is a primitive isotropic element of V and we can find some hyperbolic plane H0 that contains v and write V = H0 K W (cf. Corollary 2.1.3). Now for an arbitrary v P S˚(V) the same is true and we obtain a decomposition V = H K W 1 where v P H. Clearly, there exists an isometry H0 Ñ H that sends v0 to v. In addition, H0 and H are bq -primitive because they are regular submodules. Thus Corollary 2.1.6) assures the existence of an isometry V Ñ V sending v0 to v. 21 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS Since Rv is contained in a hyperbolic plane H = Rv + Rv 1, we split this plane from V and obtain V = H K W. For φ P O(V) to be in stabO(V)(v), it is necessary and sufficient to satisfy φ(v) = v,q(φ(v 1)) = 0,b(v,φ(v 1)) = 1. By Witt’s Theorem on the extension of isometries (cf. Corollary 2.1.6) we can choose any u P V satisfying these conditions and find φ P O(V) with φ(v) = v, φ(v 1) = u. Write u = λv+µv 1+w, then q(u) = q(w)+λµb(v, v 1) = q(w)+λµ as well as 1 = b(v,u) = b(v,µv 1) = µb(v,b 1) = µ. Putting this together we arrive at 0 = q(w) + λ. For each w P W this necessitates λ = ´q(w). Therefore, we obtain qs(n´2) distinct such u and conclude: | stabO(V)(v)| = | u P V q(u) = 0,b(v,u) = 1 ( | ¨ |O(W)| = qs(n´2) ¨ |O(W)|. PROPOSITION 2.3.8. Let V be a regular quadratic R-module of rank n and with ind(V) = m. Then s˚(V) is as follows: I) s˚(V) = q(s´1)(n´1) ¨ (qm ´ 1)(qm´1 + 1), II.1) s˚(V) = q(s´1)(n´1) ¨ (q2m ´ 1), II.2) s˚(V) = q(s´1)(n´1) ¨ (qm+1 + 1)(qm ´ 1). PROOF. From s˚(V) = |O(V)|/| stabO(V)(v)|, where v is an arbitrary primitive isotropic vector, and the above discussion we find | stabO(V)(v)| = qs(n´2) ¨ |O(W)|, where dim(W) = dim(V)´ 2 andW hyperbolic if and only if V hyperbolic. Therefore, s˚(V) = q´s(n´2) ¨ |O(V)||O(W)| = q´s(n´2) ¨ q(s´1)(2n´3) ¨ |O(V)||O(W)| = q´s(n´2) ¨ q(s´1)(2n´3) ¨ qn´2 ¨ s˚(V) = q(s´1)(n´1) ¨ s˚(V), where V ,W are the corresponding reductions modulo m. The second equality follows from Corollary 2.2.8 and the third equality follows from Fact 2.1.13. Filling in the quantities s˚(V) from Fact 2.1.12 we arrive at the claim. We proceed with s(V). We use the above introduced (cf. 2.3.1) slicing of V . PROPOSITION 2.3.9. Let V be a regular quadratic R-module of rankn and with ind(V) = m. If V is hyperbolic sk˚(V) = # (qm ´ 1)(qm´1 + 1) ¨ qkn ¨ q((s´2k)´1)(n´1) k ă s/2 q(s´k´1)n(qn ´ 1) k ě s/2, s(V) = (qm ´ 1)(qm´1 + 1) ¨ (rs/2s´1ÿ k=0 qnk ¨ q((s´2k)´1)(n´1) ) + q(s´rs/2s)n, 22 2.4. A CLASSIFICATION OF M.T.I.S. OF REGULAR QUADRATIC MODULES OVER FINITE LOCAL RINGS if V is non-hyperbolic of odd rank sk˚(V) = # (q2m ´ 1) ¨ qkn ¨ q((s´2k)´1)(n´1) k ă s/2 q(s´k´1)n(qn ´ 1) k ě s/2, s(V) = (q2m ´ 1) ¨ (rs/2s´1ÿ k=0 qnk ¨ q((s´2k)´1)(n´1) ) + q(s´rs/2s)n, and if V is non-hyperbolic of even rank sk˚(V) = # (qm+1 + 1)(qm ´ 1) ¨ qkn ¨ q((s´2k)´1)(n´1) k ă s/2 q(s´k´1)n(qn ´ 1) k ě s/2, s(V) = (qm+1 + 1)(qm ´ 1) ¨ (rs/2s´1ÿ k=0 qnk ¨ q((s´2k)´1)(n´1) ) + q(s´rs/2s)n. PROOF. We fix t := rs/2s. Write V = Ťs k=0 V ˚ (k). First of all we have seen that for 2k ě s each of the q(s´k)n distinct elements of V(k) is isotropic (cf. Lemma 2.3.5). Because of V(t) = Ť s/2ďkďs V pr (k), we write V = Ťt k=0 V pr (k)YV(t). This decomposition is disjoint, so we find s(V) = řt k=0 s ˚(Vpr(k)) + s(V(t)) = řt k=0 s ˚(Vpr(k)) + q (s´t)n. Following Lemma 2.3.5, in the case 2k ă s a vector v P V˚(k) is isotropic if and only if the corresponding v P V(k) satisfies q(v) = 0 (here we work with the reduction modulo mr´2k/mr´k). By construction of V(k), each primitive isotropic vector in V(k) represents q(s´k)n/q(s´2k)n = qkn distinct isotropic elements in Vpr(k). This states that there are q kn ¨ s˚(V(k)) isotropic vectors in Vpr(k). But V(k) is regular, as noted after its construction, so we already know the quantities s˚(V(k)) by Proposition 2.3.8. Since V(k) is hyperbolic if and only if V(k) is hyperbolic if and only if V is hyperbolic, we arrive at s(V) = s˚(V) ¨ ( t´1ÿ k=0 qkn ¨ q((s´2k)´1)(n´1) ) + q(s´t)n. 2.4 A CLASSIFICATION OF MAXIMAL TOTALLY ISOTROPIC SUBMODULES OF REGULAR QUADRATIC MODULES OVER FINITE LOCAL RINGS Throughout this Section R will be a finite local ring of cardinality s, with maximal ideal m and corre- sponding nilpotency index r, if not specified otherwise. 2.4.1 Reduction of the general case As in the case of isometry above, the problem of identifying maximal totally isotropic submodules obeys to the local-global principle of Theorem 2.2.2. 23 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS PROPOSITION 2.4.1. Let R be a finite ring with spectrum specR = tm1, . . . ,mlu. Let (V ,q) be a quadratic R-module, and letM be a submodule. Then M – lź i=1 Mmi – lź i=1 M/mi. M is (maximal) totally isotropic if and only ifMm is (maximal) totally isotropic in (Vm,qm) for all m P specR. PROOF. The decomposition of M into local factors and quotient rings is clear since it holds for all finitely generated modules over R. Since a module (M,q) is totally isotropic if and only if it is isometric to (M, 0), where 0 is the trivial quadratic form, and since (M, 0)m = (Mm, 0) for all maximal ideals, the assertion of the Proposition follows by ii. of Theorem 2.2.2. 2.4.2 A structure theorem for maximal totally isotropic submodules of regular modules over finite rings We achieve a complete classification of maximal totally isotropic submodules of regular modules over the finite ring R by a structural decomposition of any such submoduleM into blocks that are supported on a rank 2 submodule. This is the main result of this Chapter and its proof will be given in quite some detail. We fix some notation: LetM be a maximal totally isotropic submodule of V . We let νM := min νm(m) m PM : q(m) = 0 ( denote the minimal m-order of an isotropic element inM. DEFINITION 2.4.2. Let H = Re+Rf – H be a hyperbolic plane over R, where (e, f) is a hyperbolic pair. Then M = mke + mr´kf is a maximal totally isotropic submodule. We write He,f(k) = He,f(νM) for this submodule. If we are interested in a representative of an M as such up to isometry, we will use the notation H(k). ˛ EXAMPLE 2.4.3. Let (V ,q) be a regular free quadratic module. We can construct examples of maximal totally isotropic submodules from rank-two building blocks. i. Let V = H1 K . . . K Hm, with Hi = Rei + Rfi – H and (ei, fi) hyperbolic pairings. Let 0 ď k1, . . . ,km ď r r2 s be integers. Then M := H1e1,f1(k1) K . . . K Hmem,fm(km) is a maximal totally isotropic submodule. ii. Let V = H1 K . . . K Hm K W, with Hi = Rei + Rfi – H and ei, fi hyperbolic pairings, W non-hyperbolic, free regular of rank 1 or 2. Let 0 ď k1, . . . ,km ď r r2 s be integers. Then M := H1e1,f1(k1) K . . . K Hmem,fm(km) K mrr/2sW is a maximal totally isotropic submodule. We will not dwell on providing a proof, for the maximality of the submodules above, here. This will be proven along the lines of the proof of the structural decomposition Theorem in 2.4.3.  This example already contains a representative for each isometry class of maximal totally isotropic submodules of free regular quadratic modules over R. This is made precise in the main Theorem of this Chapter, which gives a structural decomposition of any maximal totally isotropic submodule, which looks as the examples above. 24 2.4. A CLASSIFICATION OF M.T.I.S. OF REGULAR QUADRATIC MODULES OVER FINITE LOCAL RINGS THEOREM 2.4.4 (STRUCTURAL DECOMPOSITION FOR MAXIMAL TOTALLY ISOTROPIC SUBMODULES). Let R be a finite local ring with maximal ideal m and corresponding nilpotency index r. Let V be a free regular quadratic module over R, with ind(V) = m. Let M be a maximal totally isotropic submodule of V . Then there exists a (unique) sequence νM = k1 ď . . . ď km ď r r2 s together with a decomposition of V into hyperbolic planes Hi = Rei + Rfi, and possibly some non-hyperbolic moduleW of rank 1 or 2, such that: I) V is hyperbolic (rk(V) = 2m) and V = H1 K . . . K Hm M = H1e1,f1(k1) K . . . K Hmem,fm(km), II) V is non-hyperbolic (rk(V) = 2m+ 1 or rk(V) = 2m+ 2) and V = H1 K . . . K Hm KW M = H1e1,f1(k1) K . . . K Hmem,fm(km) K mrr/2sW. DEFINITION 2.4.5. Let V be a free regular quadratic module over R and M Ă V be a maximal totally isotropic submodule. We define the type ofM to be the sequence [k1, . . . ,km], if V is hyperbolic, or [k1, . . . ,km, rr/2s], if V is non-hyperbolic, where the integers νM = k1 ď . . . ď km ď r r2 s are those obtained from Theorem 2.4.4. ˛ REMARK 2.4.6. It is quite clear that the notion of type as defined above is well-defined. For if M – H(k1) K . . . K H(km) KW 1 andM – H(l1) K . . . K H(lm) KW 1 for admissible sequences [k1, . . . ,km], [l1, . . . , lm], the equality [k1, . . . ,km] = [l1, . . . , lm] is inevitable. To see this consider the first index j from the left where kj ‰ lj, w.l.o.g. we assume kj ą lj. Then it is easy to see that the number of elements with m- order kj in H(k1) K . . . K H(km) K W 1 would be strictly larger than the number of such elements in H(l1) K . . . K H(lm) KW 1, while both are isometric toM. This is absurd.  The discussion of the examples in 2.4.3 and the structural decomposition Theorem 2.4.4 immediately characterizes the set of possible types of a given free regular quadratic module V over R. COROLLARY 2.4.7. Let V be a regular quadratic module over R with ind(V) = m. i) If V is hyperbolic, a sequence [ν1, . . . ,νm] is the type of a maximal totally isotropic submodule of V if and only if νi P t0, . . . , rr/2su is a non decreasing sequence. ii) If V is non-hyperbolic, a sequence [ν1, . . . ,νm, rνm+1s] is the type of a maximal totally isotropic submod- ule of V if and only if νm+1 = rr/2s and νi P t0, . . . , rr/2su is a non decreasing sequence. As mentioned before, the case of primitive maximal totally isotropic submodules over local rings is not much different from the situation over fields. In particular, their rank is an invariant (cf. Corollary 2.1.8), equal to ind(V). 25 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS COROLLARY 2.4.8. Let V be a regular quadratic module over a finite local principal ideal ring R, and letM be a maximal totally isotropic submodule of type t. i. If r is even, then |M| = qr(n/2). ii. If r is odd, then II.1) |M| = q(r´1)/2 ¨ qr(n´1/2), II.2) |M| = q(r´1) ¨ qr((n/2)´1). In particular |M| = qs(n/2) if and only if r is even or ifM is hyperbolic. Furthermore, it is an easy consequence of the fact that O(V) acts transitively on the set of primitive isotropic elements (cf. Lemma 2.3.7) that all primitive maximal totally isotropic submodules lie in one orbit under the action of O(V). This situation generalizes to non-primitive maximal totally isotropic submodules. COROLLARY 2.4.9. Let V be a regular quadratic module over R and let t be a type of V . Then for any maximal totally isotropic submodulesM,M 1 of type t there exists φ P O(V) such that φ(M) =M 1. 2.4.3 The proof of the structural decomposition Theorem We will work through the proof of the structural decomposition Theorem in several steps. We start with the description of the maximal totally isotropic submodules of the regular free non-hyperbolic and hyperbolic modules of rank 1 and 2 in Lemmata 2.4.10 and 2.4.11. The general situation will then be reduced to these. To do so, Lemma 2.4.14 provides a way to split off someH(k) from a givenM. This can be done in such a way that M – H(k) K M 1, where M 1 is a maximal totally isotropic submodule of a free regular quadratic module V 1, which is Witt-equivalent to V and satisfies rk(V 1) = rk(V)´ 2. LEMMA 2.4.10. LetW be a non-hyperbolic regular quadratic module of rank 1 or 2 over R. Then i. w PW is isotropic if and only if νm(w) ě rr/2s, that is, w P mrr/2sW. ii. mrr/2sW is the unique maximal totally isotropic submodule ofW. PROOF. It is clear that W has no primitive isotropic elements, for those would project to primitive isotropic elements of W, which itself is an anisotropic space. Along the same lines we conclude that q(w) P Rˆ for all primitive w PW; if w PW with q(w) P RzRˆ = m, then w is isotropic modulo m and therefore w is the zero element ofW, thus w cannot be primitive. If w P mkW is isotropic, we write w = µkw 1 with w 1 primitive and µ P mzm2. Then q(w) = µ2kq(w 1). Since q(w 1) P Rˆ we have νm(q(w)) = 2k. So necessarily k ě r/2 if q(w) = 0. Clearly mrr/2sW is a totally isotropic subspace and, by the above, there are no isotropic elements in Wzmrr/2sW. This completes the proof. Having settled the non-hyperbolic case of rank 1 and 2, we deal with the second basic building block, hyperbolic planes. 26 2.4. A CLASSIFICATION OF M.T.I.S. OF REGULAR QUADRATIC MODULES OVER FINITE LOCAL RINGS LEMMA 2.4.11. Let H be a hyperbolic plane over R. LetM be a maximal totally isotropic submodule ofM and let x PM be such that k := νm(x) = νM. Then there exists a hyperbolic pair (e, f) such that i. H = Re+ Rf, ii. M = He,f(k) = mke+mr´kf. PROOF. Let x 1 be primitive in H, so that x = λx 1 for λ P mk. Given x 1 (not necessarily isotropic itself), there exists some primitive isotropic element f P H, such that b(x 1, f) = 1. For if we choose any isotropic basis h1,h2 of H, non-existence of an f as proposed would mean that b(f,hi) P m, a contradiction to the regularity of H. We now show that M = mkx 1 + mr´kf. If k = 0 this is clear since every primitive isotropic element x of H is contained in a uniquely determined maximal totally isotropic submodule Rx, as in the case of a field. If k ą 0, we see that αx 1 R M for all α P Rzmk, since νm(αx 1) = νm(α) ă k = νm(M) would be a contradiction. Now the submodule Rx ĹM, because Rx is not maximal totally isotropic. In fact let α P mk, β P mr´k, then q(βf) = 0, and q(αx 1 +βf) = αβ P mr = t0u, showing Rx = mkx 1 Ĺ mkx 1 +mr´kf, where the latter submodule is totally isotropic. Therefore, there exists y PMzRx. Write y = αx 1 + βf. Since νm(M) = k, it follows that α,β P mk. But then βf = y´αx 1 PM and therefore mkx 1 +Rβf ĂM. In particular, for some µ P mzm2 we set z := µkx 1 + βf P M and obtain q(z) = µkβ = 0, showing β P mr´k. Therefore y P mkx 1 + mr´kf and thus M Ă mkx 1 + mr´kf. But since the latter is a totally isotropic subspace andM is a maximal totally isotropic subspace it follows that equality holds. We now show that there is a hyperbolic pair (e, f) such thatM = mkx 1+mr´kf = mke+mkf. Suppose that x 1 is not isotropic itself, otherwise e := x 1 and f are a basis as desired. Since x 1, f are a basis of H, so is x 1 + βf, f, for β P R. Then q(x 1 + βf) = q(x 1) + β2q(f) + β ¨ bq(x 1, f) = q(x 1) + 0 + β ¨ 1 = q(x 1) + β implies, that with e := x 1 ´ q(x 1)f, we arrive at a hyperbolic pair (e, f). We claim that λe P M. To see this note that q(x 1) P mr´2k because otherwise νm(q(x)) = νm(q(λx 1)) = νm(λ2) + νm(q(x 1)) ă 2k+ r´ 2k = r, contradicting q(x) = 0. Thus λe = λx 1´(λq(x 1))f P mkx 1+mr´kf =M, since νm(λq(x 1)) ě k+r´2k = r´ k. It now follows that mke+mr´kf = mkx 1 +mr´kf =M. This settles the problem for the rank 1 and 2 modules. We proceed by a reduction of the general situation to the rank 1 and 2 case. LEMMA 2.4.12. Let V be a regular quadratic module over R, such that rk(V) ě 3. Then there exists a basis consisting of isotropic elements. PROOF. This clearly is true if V is hyperbolic. So assume that V is not hyperbolic and write n := rk(V) and V = H1 K . . . K Hs K W, where W is a regular non-hyperbolic quadratic module of rank 1 or 2. We concatenate a basis h1, . . . ,hn´2 of the hyperbolic part and a basis of W, to obtain a basis derived from the decomposition in an obvious way. W.l.o.g. we can assume that q(hi) = 0 for all indices i. By 27 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS assumption at least one hyperbolic plane splits V , and by its universality we can, for each w in a basis of W, choose some hw P H such that q(hw) = ´q(w). Replacing w by w + hw in the above basis we arrive at a basis composed of isotropic elements. LEMMA 2.4.13. Let V be a regular quadratic module over R, such that rk(V) ě 3. Let x P V be primitive, such that λx is isotropic for some λ with νm(λ) ă r2 . Then there exists a submodule H = Rx+ Rv that splits V , such that H – H, x P H, bq(x, v) = 1, and q(v) = 0 are satisfied. PROOF. Choose some primitive isotropic v P V that satisfies bq(x, v) = 1. Such an element has to exist, since there is some isotropic basis of V by Lemma 2.4.12 and the regularity of V does not allow that bq(x, v) P m for all v in such a basis. Set H := Rx + Rv. This is a regular submodule, where the regularity can be checked by computing the determinant of H to be equal to ´1. A regular quadratic module of rank 2, which by assumption contains the isotropic element λx P H with νm(λx) ă r2 , can not be non-hyperbolic as noted in Lemma 2.4.10. Therefore, H – H and H splits V . LEMMA 2.4.14. Let V be a regular quadratic module over R, such that rk(V) ě 3. Let M be a maximal totally isotropic submodule of V . Then there exists some hyperbolic plane H Ă V , with corresponding splitting V = H K V 1, such that i. M = H 1 KM 1, where H 1 Ă H andM 1 Ă V 1. ii. H 1 =MXH is a maximal totally isotropic submodule of H and νMXH = νM, iii. M 1 =MX V 1 is a maximal totally isotropic submodule of V 1 and νMXV 1 ď νM. PROOF. Assume that νM ă r2 , because otherwise M Ă mrr/2sV , where the latter is totally isotropic, thus implying equality and the claim holds by Witt decomposition of V . If k := νM ă r2 , there is some x P M with νm(x) = νM = k ă r2 , so there exist a λ P R and a primitive x 1 P V with νm(λ) = k ă r2 such that x = λx 1. Using Lemma 2.4.13 we find H = Rx 1 + Rv Ă V hyperbolic such that x 1 P H, v is isotropic and V = H K V 1. STEP 1: We show that mkVXxK = H 1 K mkV 1, whereH 1 = mkx 1+mr´kv is a maximal totally isotropic submodule of H. Since mkV 1 Ă mkV X xK, we can write mkV X xK = mkH X xK K mkV 1. We are left to show that mkHX xK = mkx 1 +mr´kv, where the “Ą” inclusion is straightforward. Towards the “Ă” inclusion we choose y P mkH = mkx 1 + mkv with bq(x,y) = 0. We can write y = αµ1x 1 + βµ2v with µ1,µ2 P mkzmk+1, whence 0 = bq(x,y) = bq(λx 1,αµ1x 1 + βµ2v) = αbq(λx 1,µ1x 1) + λβµ2b(x 1, v) = λµ2β. Only the last equality needs some justification: by choice of x we have bq(x, x) = 2q(x) = 0. Now bq(x, x) = bq(λx 1, λx 1) and for this to equal 0 necessarily νm(bq(x 1, x 1)) ě r ´ 2νm(λ) = r ´ 2k has to hold, so bq(x 1, x 1) P mr´2k. But then also bq(λx 1,µ1x 1) = λµ1b(x 1, x 1) P mk ¨ mk ¨ mr´2k = t0u. Furthermore bq(x 1, v) = 1 by construction of H (cf. Lemma 2.4.13). Having justified this, we observe that clearly λµ1β = 0 if and only if β P mr´2k if and only if βµ1 P mr´k. Maximality ofH 1 follows from the classification made in Lemma 2.4.11. This concludes the first step. 28 2.4. A CLASSIFICATION OF M.T.I.S. OF REGULAR QUADRATIC MODULES OVER FINITE LOCAL RINGS STEP 2: We show thatM Ă H 1 K mkV 1. It is clear thatM Ă xK, and by assumptionM Ă mkV . ThusM Ă mkV X xK. By the identity mkV X xK = H 1 K mkV 1, which was proved in STEP 1, we therefore obtainM Ă H 1 K mkV 1. This concludes the second step. STEP 3: We show that H 1 ĂM. For y P H 1 we findM Ă Ry+M. We claim that Ry+M is totally isotropic, which then implies equality by the maximality of M. Take any element αy +m where m P M can be written as m = h + v with h P H 1 and v P mkV 1. Then q(αy+m) = bq(αy,m) = bq(αy,h+ v) = bq(αy,h) + bq(αy, v) = q(αy+ h) = 0, where bq(αy, v) = 0 by the orthogonality of H 1 and V 1. Furthermore, αy,h P H 1, so αy + h P H 1 and 0 = q(αy+ h) = b(αy,h) by the total isotropy of H 1. This concludes the third step. STEP 4: We show that M = H 1 K M 1, where M 1 := M X mkV 1 Ă V 1 is a maximal totally isotropic submodule of V 1. By step 2 we can write an m PM as m = h+ v with h P H 1 and v P mkV 1. By the inclusion H 1 ĂM of step 3 then also v = m´ h PM. With M 1 :=MXmkV 1 Ă V 1 we obtain M = H 1 KM 1. H 1 is maximal totally isotropic in H by step 1. Furthermore M 1 is totally isotropic with νM1 ě νM, and in fact, M 1 is a maximal totally isotropic submodule of V 1, because for totally isotropicM2 Ă V 1 withM 1 ĂM2 we have M Ă H 1 +M2, where the maximality of M forces equality and shows M 1 =M2. This concludes the fourth step. Conclusion of the proof: We just established that M = H 1 KM 1 with H 1 =MXH Ă H maximal totally isotropic by step 1 and M 1 =MX V 1 Ă V 1 maximal totally isotropic by step 5. This concludes the proof. Before we go on to put all pieces together and give a proof of Theorem 2.4.4, we formulate a Corollary to the preceding Lemma which will be of use later on (cf. Lemma 2.5.2). COROLLARY 2.4.15. Let V be a regular quadratic module over R and let M be a maximal totally isotropic submodule. Let x P M, such that νm(x) = νM and that there exists a hyperbolic plane H containing x. Then V = H K V 1 with V ,V 1 Witt-equivalent, and M = H(νM) K M 1, where M 1 is a maximal totally isotropic submodule of V 1. PROOF OF THEOREM 2.4.4. We start with the existence of a decomposition as claimed. If rk(V) ď 2 this is covered by the Lemmata 2.4.10 and 2.4.11. If rk(V) ě 3 we proceed by induction with Lemma 2.4.14: We write V = H K V 1 and see that M = M X H K M X V 1 where H 1 = M X H and M 1 = M X V 1 are maximal totally isotropic subspaces of H and V 1. By Lemma 2.4.11, H 1 = H(νM) – H(νM). Furthermore since V and V 1 are Witt-equivalent and since rk(V 1) = rk(V)´ 2 we inductively get M 1 = H2(k2) K . . . K Hm(km), or M 1 = H2(k2) K . . . K Hm(km) K mrr/2sW, 29 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS depending on V being hyperbolic or not. Uniqueness of a decomposition as such, up to isometry, was already discussed in Remark 2.4.6. 2.5 COUNTING MAXIMAL TOTALLY ISOTROPIC SUBMODULES OF QUADRATIC MODULES OVER FINITE PRINCIPAL IDEAL RINGS Let R be a finite principal ideal ring. Let M(V) denote the set of all maximal totally isotropic submod- ules of V and let M(V , t) denote the set of maximal totally isotropic submodules of V , that are of type t. We setm(V) := |M(V)| andm(V , t) := |M(V , t)|. 2.5.1 Reduction of the general case Employing the results on the decomposition of modules over finite rings, we immediately derive the following multiplicativity result. PROPOSITION 2.5.1. Let (V ,q) be a regular quadratic module over R. Then m((V ,q)) = ź mPspecR m((Vm,qm)), where m((Vm,qm)) is the number of maximal totally isotropic submodules of the regular quadratic Rm-module (Vm,qm). PROOF. This follows directly from Proposition 2.4.1. 2.5.2 The case of a finite local principal ideal ring Let R be local in addition to the assumptions already made. Let m be the maximal ideal of R. Then if |R| = qs, we find that s = r, where r is the nilpotency index of m. To compute the number of maximal totally isotropic submodules of regular quadratic modules over Rwe compute the number of maximal totally isotropic submodules of a given type, followed by summation over all different types. Let st˚1(V) denote the number of isotropic elements of V with m-order t1 = ν1, and let st˚1(t) denote the number of isotropic elements with m-order t1 in a maximal totally isotropic submodule of type t. LEMMA 2.5.2. Let V be a regular quadratic module over R and let t be a type. Let M be of type t and write V = H K V 1,M = H(t1) KM 1 as in Lemma 2.4.14. The number of maximal totally isotropic subspaces of type t obeys to the recursion m(V , t) = st˚1(V) st˚1(t) ¨m(V 1, t 1), where t 1 is the type ofM 1. PROOF. Let x P Sν˚1(V), and let H be any hyperbolic plane in V that contains x (as is given by Lemma 2.4.13) and write V = H K V 1. Let N ĂM(V , t) be composed of those maximal totally isotropic spaces of type t that contain x. By Corollary 2.4.15 we can write any M P N as M = H(ν1) KM 1 where M 1 is a maximal totally isotropic submodule of V 1. Then M 1 is of type t 1 := [t2, . . . , tn/2] by the uniqueness of type (cf. Theorem 2.4.4). So there are exactly m(V 1, t 1) distinct elements of M(V , t) that contain x. 30 2.5. COUNTING M.T.I.S. OF QUADRATIC MODULES OVER FINITE PRINCIPAL RINGS Of course m(V 1, t 1) does not depend on V 1 itself, but only on the Witt-equivalence class of V 1 and its rank. If we do so for all of the st˚1(V) elements of St˚1(V), we meet each element M P M(V , t) exactly st˚1(t) times: Once for each isotropic element x PMwith ν(x) = t1, of which there are st˚1 by definition of this quantity. But this translates tom(V , t) ¨ st˚1(t) = st˚1(V) ¨m(V 1, t 1) and therefore proves the claim. So we are left with the need to compute the quantities st˚1(t) and st˚1(V), in order to be able to compute m(V , t). The latter one is given in Proposition 2.3.9, the former one is calculated below. We fix some notation. Given a type t = [t1, . . . , tn/2] for some V with ind(V) = m, let (s1, . . . , sa) be the sequence of numbers that indicate how many consecutive entries of t are equal before each increment that does not reach r2 and let s = s1 + . . . + sa. That is, t1 = . . . = ts1 ă ts1+1 = . . . = ts1+s2 ă . . . ă ts1+...sa´1+1 = . . . = ts ă r2 . Then we have that s ď m and m 1 := rn/2s ´ s is exactly the number of entries of t equal to r r2 s. Furthermore, we set t(j) := ts1+...+sj´1+1, that is, t (j) picks out the j-th smallest value of t. REMARK 2.5.3. For a type t of a regular quadratic R-module V certain restrictions apply for the numbers s,m 1. These are: i. V hyperbolic, r odd: s = m,m 1 = 0; ii. V hyperbolic, r even: 0 ď s ď m, 0 ď m 1 ď m, such that s+m 1 = m; iii. V non-hyperbolic, r odd: s = m,m 1 = 1; iv. V non-hyperbolic, r even: 0 ď s ď m, 1 ď m 1 ď m+ 1, such that s+m 1 = m+ 1.  LEMMA 2.5.4. Let V be a regular quadratic module over R, and let t be a type of V . Then if t1 ă r r2 s: I) st˚1(t) = q rm(1´ q´s1), II.1) st˚1(t) = q rm(1´ q´s1) ¨ qtr/2u, II.2) st˚1(t) = q rm(1´ q´s1) ¨ q2tr/2u. If t1 = r r2 s, thenM = m rr/2sV and therefore st˚1(t) = s ˚ rr/2s(V). PROOF. Let M Ă V be a maximal totally isotropic subspace of type t, and let m = ind(V). We use Theorem 2.4.4 to obtain a decomposition V = H1 K . . . K Hs K V 1, where V 1 XM = mrr/2sV 1 and we write n 1 = rk(V 1) = n ´ 2s. Therefore, we can find a basis e1, f1, . . . , es, fs, v1, . . . , vn1 of V (each Rei + Rfi – H, V 1 = Rv1 ‘ . . .‘ Rvn1 ) in which M =Ms1 KMs´s1 KMm1 31 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS where Ms1 = m t(1)e1 +m r´t(1)f1 + . . . +mt(s1)es1 +mr´t(s1)fs1 , Ms´s1 = mt(s1+1)es1+1 +mr´t(s1+1)fs1+1 . . . +mt(s)es +mr´t(s)fs, Mm1 = m rr/2sV 1. For x PMwe accordingly write x = xs1 + xs´s1 + xm1 . Such x is primitive relative toM, i.e., a member of St˚1(t), if and only if xs1 is primitive relative toM, i.e., if and only if xs1 = α1µ t1 e1 e1 + β1µ r´t1 f1 f1 + . . . + αs1µ ts1 es1 e1 + βs1µ r´ts1 fs1 fs1 , where µei ,µfi P mzm2, with at least one αi P Rˆ. There is a total of q(r´1)s1 elements of Ms1 that are not primitive relative to M. This follows from the observation that for the s1 hyperbolic pairs (ei, fi) that span Ms1 we may not use a unit αi. This leaves us with q(r´1) different possibilities to choose α P RzRˆ = m. These scalars then amount to q(r´1)´t1 distinct elementsαµt1eiei for each primitive ei. To such anαi we can choose an arbitraryβ P R, giving qt1 distinct elements βµt1fifi for each primitive fi. In total, each pair (ei, fi) thus contributes to q(r´1)´t1 ¨ qt1 = qr´1 different elements ofMs1 not primitive relative toM. Since in the above decomposition of x, the summands xs´s1 and xm1 can be chosen arbitrarily and since |Ms´s1 | = qr(s´s1), |Mm1 | = qn1tr/2u, we have a total of( qrs1 ´ q(r´1)s1 ) ¨ qr(s´s1) ¨ qn1tr/2u elements in St˚1(t). Depending on the parity of r and whether n 1 = 2m 1 or n 1 = 2m 1 ´ 1 this can be simplified to the formulation in the Lemma. We now can computem(V , t) for given V and t. PROPOSITION 2.5.5. Let V be a regular quadratic module over R, with ind(V) = m. Then: if V is hyperbolic m(Hm, t) =q(r´2)(ms´s(s+1)/2)q´s ¨ sź i=1 (q(m´i+1) ´ 1)(q(m´i) + 1) ¨ s1ź i=1 qi(qi ´ 1)´1 ¨ . . . ¨ srź i=1 qi(qi ´ 1)´1 ¨ s1ź i=1 q´2(m´i)t(1) ¨ . . . ¨ sź i=s1+...+sa´1+1 q´2(m´i)t(a) , if V is non-hyperbolic of odd rank m(Hm KW, t) =q(r´2)((m+1)s´s(s+1)/2)q´str/2u ¨ sź i=1 (q2(m´i+1) ´ 1) ¨ s1ź i=1 qi(qi ´ 1)´1 ¨ . . . ¨ srź i=1 qi(qi ´ 1)´1 ¨ s1ź i=1 q´(2(m´i+1)´1)t(1) ¨ . . . ¨ sź i=s1+...+sa´1+1 q´(2(m´i+1)´1)t(a) , 32 2.5. COUNTING M.T.I.S. OF QUADRATIC MODULES OVER FINITE PRINCIPAL RINGS and if V is non-hyperbolic of even rank m(Hm KW, t) =q(r´2)((m+1)s´s(s+1)/2)q´s(r+1 mod 2) ¨ sź i=1 (q(m´i+2) + 1)(q(m´i+1) ´ 1) ¨ s1ź i=1 qi(qi ´ 1)´1 ¨ . . . ¨ srź i=1 qi(qi ´ 1)´1 ¨ s1ź i=1 q´2(m´i+1)t(1) ¨ . . . ¨ sź i=s1+...+sa´1+1 q´2(m´i+1)t(a) . PROOF. By Lemma 2.5.4 we recursively obtain m(Hm, t) = sź i=1 st˚i(H (m´i+1)) st˚i(t[i´1]) , and m(Hm KW, t) = sź i=1 st˚i(H (m´i+1) KW) st˚i(t[i´1]) , where t[j] = [tj+1, . . . , tm] is the type obtained from t by omitting the first j entries. We write s(ďl) = s1 + . . . + sl. Using Proposition 2.3.9 and Lemma 2.5.4 we get for ti = t(l) ă r2 , that is, for i P [s(ďl´1) + 1, . . . , s(ďl)]: for V hyperbolic st˚i(H m´i+1) st˚i(t[i´1]) = (qm´i+1 ´ 1)(qm´i + 1)q(r´2)(m´i)q´1 ¨ q´2(m´i)t(l)qs(ďl)´i+1(qs(ďl)´i+1 ´ 1)´1, for V non-hyperbolic of odd rank st˚i(H m´i+1 KW) st˚i(t[i´1]) = (q2m´i+1 ´ 1)q(r´2)(m´i+1)q´tr/2u ¨ q´(2(m´i+1)´1)t(l)qs(ďl)´i+1(qs(ďl)´i+1 ´ 1)´1, and for V non-hyperbolic of even rank st˚i(H m´i+1 KW) st˚i(t[i´1]) = (qm´i+2 + 1)(qm´i+1 ´ 1)q(r´2)(m´i+1)q´(r+1 mod 2) ¨ q´2(m´i+1)t(l)qs(ďl)´i+1(qs(ďl)´i+1 ´ 1)´1. Substituting these values we get for V hyperbolic m(Hm, t) = aź l=1 slź i=1 (qm´s(ďl´1)´i+1 ´ 1)(qm´s(ďl´1)´i + 1)q(r´2)(m´s(ďl´1)´i) ¨ q´1q´2(m´s(ďl´1)´i)t(l)qsl´i+1(qsl´i+1 ´ 1)´1, for V non-hyperbolic of odd rank m(Hm KW, t) = aź l=1 slź i=1 (q2(m´s(ďl´1)´i+1) ´ 1)q(r´2)(m´s(ďl´1)´i+1) ¨ q´tr/2uq´(2(m´s(ďl´1)´i+1)´1)t(l)qsl´i+1(qsl´i+1 ´ 1)´1 33 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS and for V non-hyperbolic of even rank m(Hm KW, t) = aź l=1 slź i=1 (qm´s(ďl´1)´i+2 ´ 1)(qm´s(ďl´1)´i+1 + 1)q(r´2)(m´s(ďl´1)´i+1) ¨ q´(r+1 mod 2)q´2(m´s(ďl´1)´i+1)t(l)qsl´i+1(qsl´i+1 ´ 1)´1 respectively. After some term rewriting, we finally arrive at the formulae given in the statement of the Lemma. 2.6 LOW RANK CASES OVER FINITE PRINCIPAL IDEAL RINGS For certain low rank cases we provide a formula for the number of maximal totally isotropic submod- ules of regular quadratic modules over R. The results for modules over non-local finite rings then are obtained by the multiplicativity statement of Proposition 2.5.1. We used MAPLE for the computations presented below. We only present the results for modules of even rank here. We include the hyperbolic and non- hyperbolic case (for even r) up to rank 6 and the hyperbolic case of rank 8. This selection stems from the main application of these results to the problem of counting similar sublattices of integral lattices, where only such maximal totally isotropic submodules with cardinality qrn/2 are of interest (cf. Corollary 2.4.8 and Theorem 3.3.12). All results are in particular valid for modules over Z/prZ (with q = p). We will make use of an abbreviated form of m(V , t) for explicit types. Suppose t is a type for which, in the terminology introduced before Lemma 2.5.4, we have distinct entries j1 ă . . . ă ja for which ji occurs a total of si times. We then identifym(V , t) with mj1 . . . j1loomoon s1 ... ja . . . jalooomooon sa (V). We ignore any entry of the form r r2 s thus for examplem(V , [j1, j2, j2, r 2 , r 2 ]) (for some even r) in a module of rank 5 would be encoded bymj1j2j2(V). If the module V is clear from context, we allow ourselves to drop it from the notation. 2.6.1 Rank 2 It is straightforward that the types in the hyperbolic case are of the form [j] with j ď r2 , whereas in the non-hyperbolic case there is only the type [r r2 s]. Using Proposition 2.5.5 we see m(H, [j]) = # 2¨qr´1(Φ1(q)) qr´1(q´1) = 2 j ă r2 0 j = r2 , m(A, [j]) = # 0 j ă r2 1 j = r r2 s, and therefore m(H) = r+ 1, m(A) = 1. 34 2.6. LOW RANK CASES OVER FINITE PRINCIPAL IDEAL RINGS 2.6.2 Rank 4 We handle the hyperbolic and non-hyperbolic cases separately. The hyperbolic case There are up to 4 distinct classes of types to be considered. The types are given by [j1, j2], [j1, j1] in the case of r odd and additionally [j1, r2 ], [ r 2 , r 2 ] for r even, as long as the restrictions ji ă ji+1 and ji ă r2 apply for every occurrence. For these we compute using Proposition 2.5.5 mj1j2 = 2q r´2 ¨Φ2(q)2 ¨ q´2j1 , mj1j1 = 2q r´2 ¨Φ2(q) ¨ q ¨ q´2j1 , mj1 = q r´2 ¨Φ2(q)2 ¨ q´2j1 , We notice thatmj1j2 does not depend on the value of j2. Thus for odd r and l := r´12 : m(H2) = l´1ÿ j1=0 lÿ j2=j1+1 mj1j2 + lÿ j1=0 mj1j1 . This is essentially a geometric sum, which we simplify using the symbolic engine of MAPLE to m(H2) = qr ¨ ((r+ 1)(q2 ´ 1)´ 2q)+ 2 (q´ 1)2 . (2.5) For even rwe have to add the contribution from the types which include r2 , we fix l = r 2 ´ 1: m(H2) = l´1ÿ j1=0 lÿ j2=j1+1 mj1j2 + lÿ j1=0 (mj1j1 +mj1) + 1, Explicit evaluation of the above then shows that the formula in (2.5) is also valid for even r. The non-hyperbolic case For the following we assume that r is even. There are up to 2 distinct classes of types to be considered. The types are given by [j1, r2 ] and [ r 2 , r 2 ], as long as the restriction j1 ă r2 applies for every occurrence. For these we compute using Proposition 2.5.5 mj1 = q r´2 ¨Φ4(q) ¨ q´2j1 . Thus with l := r2 ´ 1 we obtain: m(H KW) = lÿ j=0 mj1 + 1 = qr ¨Φ4(q)´ 2 q2 ´ 1 . (2.6) 2.6.3 Rank 6 We handle the hyperbolic and non-hyperbolic cases separately. 35 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS The hyperbolic case There are up to 8 distinct classes of types to be considered. The types are given by [j1, j2, j3], [j1, j2, j2], [j1, j1, j2], [j1, j1, j1] in the case of odd r and additionally [j1, j2, r2 ], [j1, j1, r 2 ], [j1, r 2 ], [ r 2 , r 2 , r 2 ] for even r, as long as the restrictions ji ă ji+1 and ji ă r2 apply for every occurrence. Applying Proposition 2.5.5, we compute mj1j2j3 = 2q 3(r´2) ¨Φ2(q)2 ¨Φ3(q) ¨Φ4(q) ¨ q´4j1 ¨ q´2j2 , mj1j2j2 = 2q 3(r´2) ¨Φ2(q) ¨Φ3(q) ¨Φ4(q) ¨ q ¨ q´4 j1 ¨ q´2 j2 , mj1j1j2 = 2q 3(r´2) ¨Φ2(q) ¨Φ3(q) ¨Φ4(q) ¨ q ¨ q´6 j1 , mj1j1j1 = 2q 3(r´2) ¨Φ2(q) ¨Φ4(q) ¨ q3 ¨ q´6 j1 , mj1j2 = q 3(r´2) ¨Φ2(q)2 ¨Φ3(q) ¨Φ4(q) ¨ q´4 j1 ¨ q´2 j2 , mj1j1 = q 3(r´2) ¨Φ2(q) ¨Φ3(q) ¨Φ4(q) ¨ q ¨ q´6 j1 , mj1 = q 2(r´2) ¨Φ3(q) ¨Φ4(q) ¨ q´4 j1 . We notice thatmj1j2j3 does not depend on the value of j3 and thatmj1j1j2 does not depend on the value of j2. Thus for odd r and l := r´12 : m(H3) = l´2ÿ j1=0 l´1ÿ j2=j1+1 lÿ j3=j2+1 mj1j2j3 + l´1ÿ j1=0 lÿ j2=j1+1 (mj1j2j2 +mj1j1j2) + lÿ j1=0 mj1j1j1 . This is essentially a cascade of geometric sums, which we simplify using the symbolic engine of MAPLE and some additional work by hand. This gives m(H3) = q3r ¨ a3r + q2r ¨ a2r + a0 (q3 ´ 1) (q2 ´ 1) (q´ 1) , (2.7) where a3r = ( (r+ 1)(q3 ´ 1)´ 2q ¨Φ2(q) ) ¨Φ2(q) ¨Φ4(q), a2r = 2 ¨Φ3(q)2, a0 = ´2. For even rwe have to add the contribution from the types which include r2 , we fix l = r 2 ´ 1: m(H3) = l´2ÿ j1=0 l´1ÿ j2=j1+1 lÿ j3=j2+1 mj1j2j3 + l´1ÿ j1=0 lÿ j2=j1+1 (mj1j2j2 +mj1j1j2 +mj1j2) + lÿ j1=0 (mj1j1j1 +mj1j1 +mj1) + 1. Explicit evaluation of the above then shows that the formula in (2.7) is also valid for even r. 36 2.6. LOW RANK CASES OVER FINITE PRINCIPAL IDEAL RINGS The non-hyperbolic case For the following we assume that r is even. There are up to 4 distinct classes of types to be considered. The types are given by [j1, j2, r2 ], [j1, j1, r 2 ], [j1, r 2 , r 2 ], [ r 2 , r 2 , r 2 ], as long as the restrictions ji ă ji+1 and ji ă r2 apply for every occurrence. Applying Proposition 2.5.5, we compute mj1,j2 = q 3(r´2) ¨ (q3 + 1) ¨Φ2(q) ¨Φ4(q) ¨ q´4 j1 ¨ q´2 j2 , mj1,j1 = q 3(r´2) ¨ (q3 + 1) ¨Φ4(q) ¨ q ¨ q´6 j1 , mj1 = q 2(r´2) ¨ (q3 + 1) ¨Φ2(q) ¨ q´4j1 . Thus with l := r2 ´ 1 we obtain: m(H KW) = l´1ÿ j1=0 lÿ j2=j1+1 mj1,j2 + lÿ j1=0 (mj1,j1 +mj1) + 1, which simplifies to m(H2 KW) = q 3r ¨ b3r + q2r ¨ b2r + b0 (q3 ´ 1)(q´ 1)(q2 + 1) , (2.8) where b3r = Φ4(q) 2 ¨Φ6(q), b2r = ´2 ¨Φ3(q) ¨Φ6(q), b0 = ´(q4 ´ 2q´ 1) ¨Φ6(q). 2.6.4 Rank 8 We restrict our investigation to the hyperbolic case. There are up to 16 distinct classes of types to be considered. The types are given by [j1, j2, j3, j4], [j1, j2, j3, j3], [j1, j2, j2, j3], [j1, j2, j2, j2], [j1, j1, j2, j3], [j1, j1, j2, j2], [j1, j1, j1, j2], [j1, j1, j1, j1] in the case of odd r and additionally [j1, j2, j3, r2 ], [j1, j2, j2, r 2 ], [j1, j1, j2, r 2 ], [j1, j1, j1, r 2 ], [j1, j2, r2 , r 2 ], [j1, j1, r 2 , r 2 ], [j1, r 2 , r 2 , r 2 ], [ r 2 , r 2 , r 2 , r 2 ] for even r, as long as the restrictions ji ă ji+1 and ji ă r2 apply for every occurrence. 37 2. M.T.I.S OF REGULAR QUADRATIC MODULES OVER FINITE RINGS Applying Proposition 2.5.5, we compute mj1j1j1j1 = 2q 6(r´2) ¨Φ2(q)2 ¨Φ4(q) ¨Φ6(q) ¨ q6 ¨ q´12j1 , mj1j1j1j2 = 2q 6(r´2) ¨Φ2(q)3 ¨Φ4(q)2 ¨Φ6(q) ¨ q3 ¨ q´12j1 , mj1j1j2j2 = 2q 6(r´2) ¨Φ2(q)2 ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q2 ¨ q´10j1 ¨ q´2j2 , mj1j1j2j3 = 2q 6(r´2) ¨Φ2(q)3 ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q ¨ q´10j1 ¨ q´2j2 , mj1j2j2j2 = 2q 6(r´2) ¨Φ2(q)3 ¨Φ4(q)2 ¨Φ6(q) ¨ q3 ¨ q´6j1 ¨ q´6j2 , mj1j2j2j3 = 2q 6(r´2) ¨Φ2(q)3 ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q ¨ q´6j1 ¨ q´6j2 , mj1j2j3j3 = 2q 6(r´2) ¨Φ2(q)3 ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q ¨ q´6j1 ¨ q´4j2 ¨ q´2j3 , mj1j2j3j4 = 2q 6(r´2) ¨Φ2(q)4 ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q´6j1 ¨ q´4j2 ¨ q´2j3 , mj1j2j3 = q 6(r´2) ¨Φ2(q)4 ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q´6j1 ¨ q´4j2 ¨ q´2j3 , mj1j2j2 = q 6(r´2) ¨Φ2(q)3 ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q ¨ q´6j1 ¨ q´6j2 , mj1j1j2 = q 6(r´2) ¨Φ2(q)3 ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q ¨ q´10j1 ¨ q´2j2 , mj1j1j1 = q 6(r´2) ¨Φ2(q)3 ¨Φ4(q)2 ¨Φ6(q) ¨ q3 ¨ q´12j1 , mj1j2 = q 5(r´2) ¨Φ2(q)2 ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q´6j1 ¨ q´4j2 , mj1j1 = q 5(r´2) ¨Φ2(q) ¨Φ3(q) ¨Φ4(q)2 ¨Φ6(q) ¨ q ¨ q´10j1 , mj1 = q 3(r´2) ¨Φ2(q)2 ¨Φ4(q) ¨Φ6(q) ¨ q´6j1 . We notice thatmj1,j2,j3,j4 does not depend on the value of j4, thatmj1,j2,j2,j3 andmj1,j1,j2,j3 do not depend on the value of j3 and thatmj1,j1,j1,j2 does not depend on the value of j2. Thus for odd r and l := r´12 : m(H4) = l´3ÿ j1=0 l´2ÿ j2=j1+1 l´1ÿ j3=j2+1 lÿ j4=j3+1 mj1j2j3j4 + l´2ÿ j1=0 l´1ÿ j2=j1+1 lÿ j3=j2+1 mj1j2j3j3 +mj1j2j2j3 +mj1j1j2j3 + l´1ÿ j1=0 lÿ j2=j1+1 mj1j2j2j2 +mj1j1j2j2 +mj1j1j1j2 + lÿ j1=0 mj1j1j1j1 . This is essentially a cascade of geometric sums, which we simplify using the symbolic engine of MAPLE and some additional work by hand. This gives m(H4) = q6ra6r + q 5ra5r + q 3ra3r + a0 (q5 ´ 1)(q3 ´ 1)2(q´ 1) , (2.9) 38 2.6. LOW RANK CASES OVER FINITE PRINCIPAL IDEAL RINGS where a6r = ((r+ 1)(q4 ´ 1)´ 2q ¨Φ3(q)) ¨Φ3(q) ¨Φ5(q) ¨Φ6(q), a5r = 2 ¨Φ3(q)3 ¨Φ4(q) ¨Φ6(q), a3r = ´2 ¨Φ4(q) ¨Φ5(q), a0 = 2. For even rwe have to add the contribution from the types which include r2 , we fix l = r 2 ´ 1: m(H4) = l´3ÿ j1=0 l´2ÿ j2=j1+1 l´1ÿ j3=j2+1 lÿ j4=j3+1 mj1j2j3j4 + l´2ÿ j1=0 l´1ÿ j2=j1+1 lÿ j3=j2+1 mj1j2j3j3 +mj1j2j2j3 +mj1j1j2j3 +mj1j2j3 + l´1ÿ j1=0 lÿ j2=j1+1 mj1j2j2j2 +mj1j1j2j2 +mj1j1j1j2 +mj1j2j2 +mj1j1j2 +mj1j2 + ( lÿ j1=0 mj1j1j1j1 +mj1j1j1 +mj1j1 +mj1 ) + 1. Explicit evaluation of the above then shows that the formula in (2.9) is also valid for even r. 39 CHAPTER 3 Similar sublattices This chapter contains some of the main results of this thesis. We use the arithmetic theory of integral lattices to relate similar sublattices of maximal integral lattices and maximal totally isotropic submod- ules of regular quadratic modules over the rings Z/cZ. We start by recalling basic facts about similarities of Euclidean lattices. We discuss elementary results about the structure of the set of all similarities and similar sublattices and introduce the arithmetic function ssl counting the number of similar sublattices by their index. These notions are generalized to include sublattices of a lattice L, not necessarily similar to L itself, but rather to any lattice in gen(L). We then recall basic features of the defined arithmetic functions, in particular, we discuss their super- multiplicativity and provide an example for the failure of multiplicativity in general. Furthermore we discuss a result of Conway, Rains and Sloane which was the first result to give necessary and sometimes sufficient conditions regarding the existence of similar sublattices of rational lattices. In 3.3.3 we introduce a novel approach to the problem of the enumeration of similar sublattices of maximal integral lattices. Here we find a bijective correspondence (cf. Lemmata 3.3.9, 3.3.10) between c-genus-similar sublattices of a lattice L and maximal totally isotropic submodules of L/cL, with a canonically induced regular quadratic form (cf. Lemmata 3.3.6 and 3.3.7), provided that L is maximal and integral and that gcd(det(L), c) if L is even, and gcd(2 det(L), c) if L is odd. This then implies the main results of this chapter, Theorem 3.3.12 which combines the results of the preceding Lemmata and of chapter 2 to show that under the above assumptions, the arithmetic count- ing function ssl is multiplicative and that its value on sublattices of index m = cn/2 is given by the number of maximal totally isotropic submodules of L/cL with a suitable quadratic form. In addition, we obtain Theorem 3.3.13, the analog to Theorem 3.3.12 for genus similar sublattices. We conclude this chapter with 3.4, a short discussion of Dirichlet series and Zeta functions for the cases where the above defined arithmetic function sslg is multiplicative. The lattice E8 satisfies sslE8 = ssl g E8 and is an important special case of the general result. We provide the Dirichlet series for E8 and show that it satisfies a local functional equation. 41 3. SIMILAR SUBLATTICES 3.1 SIMILAR SUBLATTICES OF LATTICES Let c P Rą0. A c-similarity, or similarity of norm c, of (V ,b) is a linear map σ : V Ñ V satisfying b(σ(x),σ(y)) = cb(x,y) for all x,y P V . We write Σ(V ,b) for the set of all similarities of (V ,b). Any similarity is bijective by the above formula and the non-degeneracy of b. Clearly, Σ(V ,b) is a subgroup of GL(V). Furthermore, the norm map N : Σ(V ,b) Ñ Rą0; σ ÞÑ c associating to a similarity σ its norm is a surjective (cf. Section 3.3.) homomorphism of groups with kernel O(V ,b). In fact if σ,σ 1 are c-similarities then there exists φ,φ 1 P O(V) such that σ = φ ˝σ 1 = σ 1 ˝φ. In particular, the set Σ(V ,b; c) := tσ P Σ(V ,b) | N(σ) = c)u, the fiber of the value c under the norm map N, is a right and left coset of O(V ,b). DEFINITION 3.1.1. Let (L,b) be a Euclidean lattice. A c-similarity, or similarity of norm c, of L, is a c-similarity σ of (RL,b) such that σ(L) Ă L. ˛ We introduce notation for the set of similarities and similar sublattices of a Euclidean lattice. DEFINITION 3.1.2. Let (L,b) be a Euclidean lattice. i. The set of similarities is Σ(L,b) := tσ P Σ(RL,b) | σ(L) = Lu. The index of a similarity σ is the index [L : σ(L)]. ii. A sublattice L 1 Ă L is a (self)-similar sublattice, short SSL, if there exists a similarity σ such that L 1 = σ(L). The norm of L 1 is the norm of σ. We set SSL(L,b) := tσ(L) Ă L | σ P Σ(L,b)u. ˛ We will subsequently count similarities and similar sublattices in terms of their index, this will align nicely into some existing framework regarding Zeta functions. Note that rewriting the counting pro- cess in terms of their norms is easily done, as we will see now. LEMMA 3.1.3. Let σ be a c-similarity of L and write n = dim(L), then [L : σ(L)] = cn/2. We define two counting functions. DEFINITION 3.1.4. Let (L,b) be a Euclidean lattice. We associate to (L,b) the similarity counting function s(L,b) : NÑ N; a ÞÑ |Σ(L,b;a)|. where Σ(L,b;a) := tσ P Σ(L,b) | [L : σ(L)] = au, and the similar sublattice counting function ssl(L,b) : NÑ N; a ÞÑ | SSL(L,b;a)|, 42 3.1. SIMILAR SUBLATTICES OF LATTICES where SSL(L,b;a) := tL 1 P SSL(L,b) | [L : L 1] = au. ˛ For the notations introduced above we will as usual sometimes abbreviate (L,b) to L if the inner prod- uct is clear from context. Having notation and vocabulary at hand, we now delve into the discussion of the most basic properties and connections of the objects in question. We first discuss invariance under scaling of a lattice. LEMMA 3.1.5. Let (L,b) be a Euclidean lattice. Let λ P Rą0. Then i. Σ(L,b) = Σ(L, λb); ii. Σ(L,b;a) = Σ(L, λb;a); iii. s(L,b) = s(L,λb); iv. ssl(L,b) = ssl(L,λb). That is, the occurring norms and indexes of similarities and similar sublattices, as well as the number of such, do not change under scaling. We collect some facts on the structure of the sets of similarities and and similar sublattices. LEMMA 3.1.6. Let (L,b) be a Euclidean lattice. i. Σ(L) Ă Σ(RL) is a monoid, but not a group. ii. O(L) = Σ(L; 1) Ă Σ(L). iii. Σ(L; c) is finite for all c P SL. PROOF. The claims are quite obvious. Towards (i) we have seen that there always is a similarity of norm greater than 1, and the inverse of this map can clearly not map into L. (ii) is clear by definition and towards (iii) we can apply a standard proof for the finiteness of O(L), which of course is a special case of the claim. The idea is to fix a basis B of L. Let λ := maxtb(v, v) | v P Bu. Then if σ is a c-similarity, σ can map the elements of B only to elements in LX Bcλ(0), those elements in L of length at most cλ. But since L is discrete and a ball is compact, this set is finite. Thus there can only be finitely many c-similarities. The sets Σ(L) (resp. Σ(L; c)) and SSL(L) (resp. SSL(L; c)) are connected to each other through an action by the orthogonal group of the lattice L: LEMMA 3.1.7. Let (L,b) be a Euclidean lattice. For the right action of O(L) on the sets Σ(L), Σ(L; c), for c P SL we obtain: i. If similarities σ,σ 1 P Σ(L) are in the same O(L)-orbit, their norms agree. 43 3. SIMILAR SUBLATTICES ii. Similarities σ,σ 1 P Σ(L; c) are in the same O(L)-orbit if and only if σ(L) = σ 1(L). iii. The O(L)-orbits on Σ(L) are of the form Σ(L)L1 = tσ P Σ(L) | σ(L) = L 1u, where L 1 P SSL(L). Thus SSL(L) is a set of representatives1 of the above action and in particular SSL(L; c) is a set of representatives for the O(L)-action on Σ(L; c). iv. sL(c) = sslL(c) ¨ |O(L)|. v. There exist r := sslL(c) distinct c-similarities σ1, . . . ,σr such that Σ(L; c) = ď˙r i=1 σiO(L). PROOF. (i) is clear. Towards (ii) the assumption σ(L) = σ 1(L) implies that σ´1 ˝ σ 1(L) = L and thus σ 1 = σ ˝ φ for some φ P O(L). On the other hand if such φ exists, clearly σ 1(L) = σ ˝ φ(L) = σ(L). The remaining claims follow immediately. 3.2 GENUS-SIMILAR SUBLATTICES OF LATTICES All of this generalizes to sublattices of a given lattice L that are not necessarily similar to L, but similar to a lattice in the genus of L. Though this may seem artificial at the moment, it is the natural environ- ment of the arithmetic method which we apply to the problem of enumerating similar sublattices. This point of view is, in particular, natural in the theory of representations of a lattice by another one. In the framework of the arithmetic theory of quadratic forms (with the notation of Chapter X in [Kne02]), the numbers sslL(cn/2) and ssl g L(c n/2) can be related to the representation numbers a(L, c ´1 L) O(L) and kÿ i=1 a(Mi, c ´1 L) O(Mi) , where tM1, . . . ,Mku is a set of representatives of gen(L). DEFINITION 3.2.1. Let (L,b) be a Euclidean lattice and let M1, . . . ,Mh Ă RL be a set of represen- tatives for the (Q-)isometry classes in gen(L). i. The set of genus similarities is Σg(L,b) := tσ P Σ(RL,b) | σ(M) = L forM P tM1, . . . ,Mhuu. The index of a similarity σ is the index [L : σ(L)]. ii. A sublattice L 1 Ă L is a genus-similar sublattice, short GSSL, if there exists a similarity σ such that L 1 = σ(M) for some M P gen(L). We say that the norm of L 1 is the norm of σ. We set SSLg(L,b) := tσ(L) | σ P Σg(RL,b)u. ˛ DEFINITION 3.2.2. Let (L,b) be a Euclidean lattice. We associate to (L,b) the genus similarity counting function sg(L,b) : NÑ N; a ÞÑ |Σg(L,b;a)|. 1We allow to refer to any set, that is in a one-to-one correspondence with the equivalence classes of some equivalence relation, as a set of representatives, rather than considering only subsets of the set the relation is defined on. 44 3.3. EXISTENCE AND ENUMERATION OF SIMILAR SUBLATTICES where Σg(L,b;a) := tσ P Σg(L,b) | [L : σ(L)] = au, and the similar sublattice counting function sslg(L,b) : NÑ N; a ÞÑ | SSLg(L,b;a)|, where SSLg(L,b;a) := tL 1 P SSLg(L,b) | [L : L 1] = au. ˛ The choice of which representatives M1, . . . ,Mh we choose does affect the above definitions of Σg(L) and Σg(L; c), but only in the following sense: If M 11, . . . ,M 1h is another set of representatives the asso- ciated sets would be in bijection, induced by the isometries ofM1 –M 11, . . . ,Mh –M 1h (which we can assume by appropriate choice of labeling). However, the function sgL is well-defined. In any case, as we make the transition to the associated sets of sublattices of L this ambiguity vanishes entirely. Now the results of Lemma 3.1.7 do not generalize directly to the case of genus-similar sublattices. This is for the fact that now we cannot let O(L) act from the right, since in general O(L) will not stabilize another lattice in the genus of L. We can however let each O(M) act on the right on those similarities σ P Σg(L) that map M into L. But this will only show that |Σg(L)| = řhi=1 ai|O(M)|, where we let ai denote the number of elements of Σg(L; c) that map M into L (in accordance with the discussion above, ai = a(M,c ´1 L) O(M) ). 3.3 EXISTENCE AND ENUMERATION OF SIMILAR SUBLATTICES 3.3.1 Properties of the counting function ssl Clearly, ssl(L,b)(1) = 1, it thus is an arithmetic function and we should ask whether it is multiplicative or not. It is not always multiplicative as the following example shows. EXAMPLE 3.3.1. On page 1392 of [BHM08] the lattice L = 2e1Z+3e2Z, together with the standard inner product, is named as an example for the failure of multiplicativity . Explicitly the first non- trivial norms of similar sublattices are 4 and 9, and these norms provide a counterexample by sslL(36) = 2 ą 1 ¨ 1 = sslL(4) ¨ sslL(9).  PROPOSITION 3.3.2. Let (L,b) be a Euclidean lattice. ssl(L,b) and ssl g (L,b) are super-multiplicative, that is, for a,a 1 coprime ssl(L,b)(aa 1) ě ssl(L,b)(a) ssl(L,b)(a 1), sslg(L,b)(aa 1) ě sslg(L,b)(a) sslg(L,b)(a 1). A proof of this can for example be found in Theorem 2.6 of [Heu10], we present the proof for conve- nience. 45 3. SIMILAR SUBLATTICES PROOF. Let M1,M2 be genus-similar sublattices of index a of L and let N1,N2 be genus similar sub- lattices of index a 1 of M1,M2 respectively. In particular N1,N2 are genus-similar sublattices of index aa 1 of L. If we assert that N1 = N2 =: NñM1 =M2, the Proposition is proven. In order to do so, we write a 1 = [Mi : N] = [Mi :M1 XM2] ¨ [M1 XM2 : N], a = [L :Mi] = [L :M1 +M2] ¨ [M1 +M2 :Mi]. Since then [M1 +M2 :Mi] = [Mi :M1 XM2]. is a common divisor of a and a 1, this index equals 1. But this showsM1 =M2. 3.3.2 Similarities of rational quadratic spaces and an existence theorem for similarities of rational lattices In the case of a Euclidean space the question of existence and even a full description of all similarities of a given norm c has a straightforward answer. For any c P Rą0 the map σc : V Ñ V ; x ÞÑ ?cx clearly is a similarity of norm c and thus the norm map is surjective. The set of all such similarities is then given as an O(V)-coset Σ(V ; c) = σcO(V). The situation already becomes more difficult by considering rational spaces with an inner product. Let (V ,b) be a positive definite quadratic Q-space. All of the above definitions for real spaces can be applied to this situation also, where clearly the codomain of the norm map then is Qą0. It is still true that the fibers of the norm map are cosets of O(V), but surjectivity can fail. PROPOSITION 3.3.3. Let (V ,b) be quadratic Q-space of dimension n. Then N(Σ(V ,b)) = tc P Qą0 | (c, (´1)n(n+1)/2dV)p = 1 for all primes p | 2c ¨ dVu. PROOF. Observe that c P N(Σ(V ,b)) if and only if cV – V since any such isometryφ : cV Ñ V satisfies cb(x,y) = b(φ(x),φ(y)) @x,y P V , and thus the underlying linear map φ : V Ñ V is a c-similarity. But now V – cV if and only if Vp – cVp for all primes p P Z, by the Hasse-Minkowski Theorem (cf. 66 : 4 in [O’M73] or the discussion in 1.3.3). We can dismiss the real spot since scaling by a positive number clearly does not affect the signature. Thus we have to check the Hasse-symbol of both spaces. These are related by (cf. p. 167 in [O’M73]): Sp cV = (c, (´1)n(n+1)/2 ¨ dVn+1)pSpV . If p is odd and does not divide either c or dV both arguments are units and thus the symbol computes to 1 in any case (cf. 63 : 12 in [O’M73]). Thus we are left to check that it equals 1 in the remaining cases, and the thus set of c P Qą0 for which this is the case is precisely the set of norms of similarities. 46 3.3. EXISTENCE AND ENUMERATION OF SIMILAR SUBLATTICES This proof is one half of the proof of the next Theorem, and was given by Conway, Rains and Sloane in their 1999 work [CRS99]. THEOREM 3.3.4 (THEOREM 1 [CRS99]). A necessary condition for a rational 2k-dimensional lattice L to have a similar sublattice of norm c is that the Hilbert symbol( c, (´1)k detL) p = 1 for all primes p dividing 2cdetL. If L is unigeneric and rZ-maximal for some r P Q then this condition is also sufficient. PROOF. The necessity of the condition is covered by Proposition 3.3.3. On the other hand if c satisfies the condition in the Theorem, the lattice ? cL is rationally equivalent to L (Q ? cL – QL) and thus there is an isometry φ of the spaces. Furthermore, φ( ? cL) is contained in an rZ-maximal lattice and there is only one such genus on each space (cf. 102 : 3 in [O’M73]). Since L is assumed to be rZ-maximal and unigeneric φ( ? cL) is a sublattice of L and it is by construction c-similar to L. 3.3.3 Enumeration of similar sublattices of integral lattices using finite quadratic modules Lemma 3.1.5 showed that the of enumeration of similar sublattices of Euclidean lattice does not depend on how that lattice is scaled. Therefore it suffices to consider integral lattices instead of rational lattices. Furthermore, we can assume that s(L) = Z. LEMMA 3.3.5. Let (L,b) be an integral lattice. Then the norm of any similarity, or similar sublattice, is an integer. PROOF. Without loss of generality we assume that s(L) = Z. Let σ be a similarity of norm c. Then [L : σ(L)] = cn/2 P N by Lemma 3.1.3. By assumption there are x,y P Lwith b(x,y) = 1. Then c = c ¨ b(x,y) = b(σ(x),σ(y)) P Z, since σ(L) is integral as a sublattice of an integral lattice. The above allows to argue for the counting functions to be indexed by the norm rather than the index. However, we stick with the way we introduced them, as functions of the index. The proof, as well as the condition, of Theorem 3.3.4 clearly has a local flavor. In the subsequent we will describe a method which relates such sublattices to maximal totally isotropic submodules of free regular quadratic modules over rings of the form Z/prZ, with p a prime. This method is subject to two constraints: It restricts to similarity factors c which satisfy gcd(c, det(L)) = 1 and enumerates sublattices which are similar to a lattice in the genus of L, rather than to L itself. If, however, L is unigeneric, we get sublattices necessarily similar to L. In addition we can show that sslL and ssl g L behave multiplicatively if the above constraints are satisfied. This method was developed by Rudolf Scharlau, the subsequent discussion is based on a manuscript [Sch11a], which was made available to the author. Publicly available slides of a talk given on this subject can be found online [Sch11b]. In particular, the subsequent results include the previously unsettled case of the root lattice E8. In addition, for all similarity factors coprime to the respective determinants we can reproduce results 47 3. SIMILAR SUBLATTICES about the root latticesA4,D4 (cf. [BM99], [BHM08]); it should be noted that the aforementioned results include the cases of similarity factors that are not coprime to the respective determinants and thus are of greater generality for these specific cases. Given an integral lattice (L,b). We define a quadratic form q as follows: if L is even, set q(x) := 12b(x, x) for all x P L, otherwise set q(x) := b(x, x) for all x P L. In any case we obtain a quadratic Z-module with values in Z. Let c P N and let denote reduction modulo c, where we assume that the modulus is clear from context. Furthermore, let pi : LÑ L/cL be the canonical projection. Then we associate to the quadratic Z-module (L,q) the free finite quadratic Z/cZ - module (L/cL,q), where q : L/cLÑ Z/cZ is the quadratic form induced by q. LEMMA 3.3.6. Let (L,b) be an integral lattice. (L/cL,q) is a regular quadratic Z/cZ-module if gcd(c, det(L)) = 1 if L is even or gcd(c, 2 det(L)) = 1 if L is odd. PROOF. In the case that L is even, q(x) = 12b(x, x), thus bq = b and the claim follows since det(L) P Z/cZ˚ is a representative for d(L/cL,q) = d(L/cL,bq) and we assumed gcd(c, det(L)) = 1. In the case that L is odd, q(x) = b(x, x), thus bq = 2b. But det(L,b) P Z/cZ˚ if and only if det(L, 2b) P Z/cZ˚,by the additional assumption 2 - c. We use this quadratic module to characterize sublattices that are c-similar to lattices in the genus of L. LEMMA 3.3.7. Let L be an integral lattice, σ be a similarity of norm c, and M = σ(L 1) be a sublattice similar to some lattice L 1 P gen(L). If gcd(c, det(L)) = 1, then cL ĂM and thereforeM/cL Ă L/cL. PROOF. By assuming gcd(c, det(L)) = 1 and the fact that [L :M] = cn/2 it follows that M# =Mc n,# X Mdet(L),#, since det(M) = [L :M]2 det(L) = cn det(L) (cf. (1.1)). We claim that Mc n,# = Mc,# = 1 c M and that L/M Ă 1 c M/M. Then cL/M Ă M/M, or equivalently, cL ĂM holds. To prove the claimed equality Mc n,# = Mc,# = 1 c M, we make a comparison of orders: Clearly, |Mcn,#| = cn by the above. Furthermore, Mc,# Ă Mcn,# by definition. Now Mc,# = 1 c M X M# and since M = σ(L 1), we find b(x,y) = b(σ(x 1),σ(y 1)) = c ¨ b(x 1,y 1) P cZ with x 1,y 1 P L 1, for arbitrary x,y P M. But then b( 1 c x,y) P Z, or equivalently 1 c x P M#. Thus Mc,# = 1 c M. Finally, | 1 c M/M| = |M/cM| = cn = |Mcn,#| and this completes the proof of the claim. If the condition gcd(c, det(L)) = 1 is violated, this can be wrong: EXAMPLE 3.3.8. Using MAGMA, we can compute similar sublattices via backtracking. We found a similarity σ of norm 3 of the Coxeter-Todd lattice K12, for which 3K12 Ć σ(K12). However, 9K12 Ă σ(K12).  LEMMA 3.3.9. Let L be an integral lattice and let c P N with gcd(c, det(L)) = 1. If M Ă L is c-similar to some lattice in the genus of L, then pi(M) Ă L/cL is a maximal totally isotropic submodule of cardinality cn/2 of the free regular quadratic module (L/cL,q). 48 3.3. EXISTENCE AND ENUMERATION OF SIMILAR SUBLATTICES PROOF. Let M Ă L be a similar sublattice of norm c. By Lemma 3.3.7 M := pi(M) =M/cL Ă L/cL is a submodule of the free regular quadratic module (L/cL,q). Since q(x) is divisible by c for all x PM, we have q(M) = t0u, thusM is totally isotropic. Furthermore, |M| = [L :M] = cn/2. With the classification of maximal totally isotropic submodules of free regular quadratic modules over Z/prZwe deduce that M is maximal (cf Corollary 2.4.8). LEMMA 3.3.10. Let L be a rZ-maximal integral lattice, and let c P N be such that gcd(c, 2 det(L)) = 1 if L is odd and gcd(c, det(L)) = 1 if L is even. If c(QL) – QL, then for every totally isotropic M Ă (L/cL,q) with |M| = cn/2 the sublatticeM := pi´1(M) is c-similar to a lattice in the genus of L. PROOF. If M Ă L/cL is a totally isotropic submodule of cardinality cn/2. M = pi´1(M) is integral and b(M,M) Ă cZ since b(x,y) = q(x+ y)´ q(x)´ q(y) P cZ for all x,y PM by isotropy of M. The lattice M 1 := c ´1 M is on c ´1 QL – cQL – QL, since M is on QL. This implies that we can replace M 1 by some lattice on the same space as L, in particular, this holds for all localizations. However, by abuse of language, we stick to the name M 1. Since there is a c-similarity from M 1 to M, we are left to show thatM 1 P gen(L). We show that there is an isometry M 1p – Lp at every prime p, which suffices because L and M 1 are positive definite. We distinguish 2 cases: i. p - c, which implies that c P Zp˚; ii. p | c, which implies that p - det(L). Ad i.: WriteM = TLwhere T is the representation matrix of a base change from L toM. T is integral by definition and det(T) = [L :M] = cn/2. Therefore Tp is in GLn(Zp), since c P Zp˚. Thus Mp = TpLp = Lp. Then L is rZ-maximal ñ Lp is rZp-maximal ñMp is rZp-maximal ñM 1p = c´1Mp is c´1rZp-maximal ñM 1p is rZp-maximal, where for the last implication we again use c P Zp˚ (cf. 1.3.3). M 1p and Lp are on the same Qp-space and up to isometry there is only one rZp-maximal lattice on this space (cf. 1.3.5, or 91 : 2 in [O’M73]). Thus Lp –M 1p, for all p - c. Ad ii.: Since det(L) = det(M 1) and p - det(L), we obtain that Lp,M 1p are both unimodular. But by assumption cQL – QL, thus we can assume thatM 1 is on QL as well. As noted before, Lp, M 1p are unimodular lattices on the Qp-space QpL. For p ‰ 2 we are in the non- dyadic case, this directly implies Lp –M 1p (cf. 1.3.4, or 92 : 2b. in [O’M73]). If p = 2, the dyadic case, L is even, since we assumed 2 - c for odd lattices. Thus, to obtain an isometry, we have to assure for Lp andM 1p to have identical norm groups (cf. 1.3.4, or Theorem 93 : 16 in [O’M73]). Let Q(x) := b(x, x) = 2q(x) for all x P L. Then g(L2) = Q(L2) + 2sLp. Now Z2 Ă g(L2) Ă 2Z2 because L2 is unimodular. Exactly one equality holds: g(L2) = Q(L2) + 2sL = 2Z2, since Q(L) Ă 2Z (L is 49 3. SIMILAR SUBLATTICES even) implies Q(L2) Ă 2Z2. All of this shows, that g(L2) = 2Z2 for all even lattices L, such that L2 is unimodular. ButM 1 is also even: q(x) = 12b(x, x) P cZ for all x PM by construction, thusQ(M) Ă 2cZ and after rescaling, c´1Q(M) Ă 2Z implies that M 1 is even. Therefore the same reasoning as above gives g(M 12) = 2Z2 = g(L2). Thus Lp –M 1p, for all p | c. We conclude: Lp –M 1p for all primes p and p =∞, thusM 1 P gen(L). This culminates in another criterion for the existence of c-similar sublattices of an integral lattice (cf. Theorem 3.3.4). Recall that in the case of odd dimension c has to be a square and that for all squares c a similar sublattice of norm c exists. We thus focus on even dimensional lattices. THEOREM 3.3.11. Let L be maximal integral lattice of rank n = 2k and let c P N such that c = śti=1 prii is the factorization of c in pairwise distinct primes pi. A necessary condition for L to have a sublattice M that is c-similar to L, is that the Fpi-space (L/piL,q) is hyperbolic for all pi, for which ri is odd. PROOF. This follows from Lemma 3.3.10 together with Corollary 2.4.8. Putting all of the above together we arrive at the main result of this section. THEOREM 3.3.12. Let L be an unigeneric maximal integral lattice and c P N such that gcd(c, 2 det(L)) = 1 if L is odd and gcd(c, det(L)) = 1 if L is even. If there is at least one similar sublattice of L with norm c, there is a one-to-one correspondence between sublatticesM that are c-similar to L and maximal totally isotropic submodules of (L/cL,q). The function sslL is multiplicative on all c satisfying the above conditions. PROOF. Lemmata 3.3.9 and 3.3.10 provide the one-to-one correspondence. Multiplicativity follows by Proposition 2.5.1. This can be slightly generalized without altering the prove, the previous Lemmata have already been formulated in accordance. THEOREM 3.3.13. Let L be an integral maximal lattice and c P N such that gcd(c, 2 det(L)) = 1 if L is odd and gcd(c, det(L)) = 1 if L is even. If there is at least one sublattice of L which is c-similar to a lattice in gen(L), there is a one-to-one correspondence between sublattices M which are c-similar to a lattice in gen(L) and maximal totally isotropic submodules of (L/cL,q). The function sslgL is multiplicative on all c satisfying the above conditions. REMARK 3.3.14. The above discussion shows one more thing. If L and c fulfill the assumptions of Theorem 3.3.12 or Theorem 3.3.13, we cannot only count (genus-)similar sublattices of norm c of L, but we can compute them explicitly using the results of Chapter 2, for example with MAGMA. To do so we could compute the orthogonal group O(L/cL,q), compute a representation of (L/cL,q) in a basis according to a Witt decomposition as in Proposition 2.2.5, and compute the orbits of O(L/cL,q) on a representative set of maximal totally isotropic submodules as in Example 2.4.3. Then Theorem 2.4.4 and Corollary 2.4.9 assure that we have computed all such submodules. It is now a standard problem, which MAGMA is capable of intrinsically, to convert these submodules of L/cL to sublattices. At the time of writing this thesis, MAGMA unfortunately does not handle quadratic modules over arbitrary finite rings, or even local finite rings of the form Z/prZ, intrinsically. A package to 50 3.4. A GLANCE TOWARDS ZETA FUNCTIONS FOR COUNTING (GENUS-)SIMILAR SUBLATTICES overcome this gap, recall that we depend on the orthogonal group of modules over such rings in the above approach, is in preparation and planned to be made available by the author.  3.4 A GLANCE TOWARDS ZETA FUNCTIONS FOR COUNTING (GENUS-)SIMILAR SUBLATTICES Let L = (L,b) be a Euclidean lattice for which we assume that sslL, or ssl g L are multiplicative. By the above results this is true for sslE8 , since E8 is unigeneric, and more generally for ssl g L for all even unimodular lattices. In this case it makes sense to define a Dirichlet series as a generating function of sslL, or ssl g L: DL(s) := ∞ÿ m=1 sslL(m) ms = ÿ L1PSSL(L) 1 [L : L 1]s , (3.1) DgL(s) := ∞ÿ m=1 sslgL(m) ms = ÿ L1PSSLg(L) 1 [L : L 1]s . (3.2) Examples of Dirichlet series related to similar sublattices of certain low dimensional lattices can be found in [BM99], [BHM08], and [BSZ11]. In the latter formulation of (3.1) and (3.2) we see a very general way of assigning Zeta functions to algebraic structures. Particular well known cases are Zeta functions of algebraic number fields, where for such a field K ζK(s) = ÿ aďoK 1 N(a)s = ÿ aďoK 1 [oK : a]s . This makes sense since the norm of ideals is multiplicative for Dedekind rings, and this can be rewrit- ten in the form ζK(s) = ∞ÿ m=1 am ms , where am = |ta ď oK | N(a) = mu. This is also true of sslL and ssl g L if these functions are multiplicative. We treat these Dirichlet series as formal sums and do not dwell on questions regarding their convergence here. By the multiplicativity result of the Theorems 3.3.12 and 3.3.13 we write DL(s) = ∞ź p ∞ÿ r=0 sslL(pr) prs , (3.3) DgL(s) = ∞ź p ∞ÿ r=0 sslgL(p r) (pr)s . (3.4) For sslgL we can use the results of 2.6 to make this decomposition into Euler factors more explicit. 51 3. SIMILAR SUBLATTICES EXAMPLE 3.4.1 (A ZETA FUNCTION FOR SIMILAR SUBLATTICES OF E8). The (even) root lattice E8 has similar sublattices of arbitrary norm c P N (cf. Theorem 3.3.4). The index of a similar sublattice of norm c is given by c4 by Lemma 3.1.3. With Theorem 3.3.12 we find sslE8(c 4) = m(H4), where H4 is the hyperbolic module of rank 8 over the appropriate ring Z/cZ. If we apply this to the Dirichlet series (3.3) we find ∞ÿ r=0 sslgL(p 4r) (p4r)s = W(p,p´s) (1´ p´s)(1 + p´s)(1 + p´2s)(1´ p3´4s)(1´ p5´4s)(1´ p3´2s)2(1 + p3´2s)2 , where W(X, Y) = (X3Y4 + 1)(X5Y2 + X5Y + 2X4Y + 2X3Y + 2X2Y + 2XY + Y + 1) = X8Y12 + X8Y8 + 2X7Y8 + 2X6Y8 + 3X5Y8 + 2X4Y8 + X3Y8 + X5Y4 + 2X4Y4 + 3X3Y4 + 2X2Y4 + 2XY4 + Y4 + 1. Note that the factors in the denominator belong to the Euler factors of zeta functions or their quotients (cf. the example to Theorem 11.7 in [Apo13]): ζ(s) = ź p 1 1´ p´s , ζ(2s) ζ(s) = ź p 1 1 + p´s . If we write Z(s) := ś pW(p,p ´s) we therefore can write DE8(s) = ζ(4s) ¨ ζ(4s´ 3) ¨ ζ(4s´ 5) ¨ ζ(4s´ 6)2 ¨ Z(s). An interesting observation is, that the Euler factors satisfy a functional equation. Write DE8,p for the complete Euler factor of DE8 at the prime p. Then DE8,p|pÑp´1 = ´p12´8s ¨DE8p. (3.5)  Observations like (3.5) are a common motive in the theory of Zeta functions of groups and rings, the interested reader is referred to [DSW08] for information on this topic. We will only add the closing remark that it should be of interest to fit the Dirichlet series obtained from the (genus)-similar sublattice counting functions into the more general framework of Zeta functions of groups and rings, the above cited book deals in some detail with Zeta functions of rings that are not necessarily associative, but additively isomorphic to Zn, just as lattices are. 52 PART II A LOCALLY EXPLICIT FORMULA FOR THE QUANTIZER CONSTANT IN MONOHEDRAL PERIODIC VECTOR QUANTIZATION 53 CHAPTER 4 Geometric foreplay This introductory chapter starts with 4.1, a collection of some basic notions about lattices and periodic point sets in Euclidean space. Here we exclude the contents of Chapter 1 from the discussion, wherever they can be spared. This is concluded by a discussion of suitable parameter spaces for lattices and periodic sets. In 4.2 we recall some facts and notions about polyhedral complexes and triangulations. Finally 4.3 covers Dirichlet-Voronoi and Delone subdivisions induced by Delone sets, in particular, by lattices and periodic sets. Duality of those polytopal complexes is discussed with the help of the more abstract notion of Dirichlet-Voronoi polytopes in dual space. 4.1 LATTICES AND PERIODIC SETS Recall from 1.3 that a (Z-)lattice L inRn is a finitely generatedZ-submodule ofRn. So L Ă Rn is a lattice in Rn if and only if there are linearly independent vectors v1, . . . , vm of Rn such that L –Àmi=1 viZ. If L is on Rn with basis B = (v1, . . . , vn) and if we write B for the matrix whose columns are given by v1, . . . , vn (in that order) we obtain L = BZn, and in this way any B P GLn(R) determines a lattice on Rn, a matrix as such is called basis matrix of the lattice L. The lattice Zn is called the standard lattice on Rn. Let L be on Rn and let B = (v1, . . . , vn) be a lattice basis of L. The fundamental parallelotope FB(L) = # nÿ i=1 λivi | λi P [0, 1], for i P t1, . . . ,nu + is a fundamental domain of L. That is, int(x + FB(L)) X int(y + FB(L)) = H for any x,y P L andŤ fPFB(L) f+ L covers R n. A periodic set Λ in Rn is a finite union of translates of a lattice L in Rn, thus it is a set of the form Λ = Ťm i=1 ti+ L, for some t = (t1, . . . , tm) P (Rn)m. We say that L is a translation lattice for Λ and that 55 4. GEOMETRIC FOREPLAY Λ is m-periodic for L, or an m-periodic set for L, if none of t1, . . . , tm is a translate of another by some element of L, that is, if the above union is disjoint. We say that m is the period of Λ for L. If L = Zn and t = (t1, . . . , tm) P (Rn)m we write Λt := Ťni=1 ti + Zn and refer to this as a standard periodic set. In particular lattices are special cases of periodic sets. If Λ is a periodic set, there always exists a unique maximal translation lattice LΛ = tv P Rn | @ x P Λ : x+ v P Λu . LetmΛ be denote the period ofΛ for LΛ, then it is the unique smallest period ofΛ, the minimal period of Λ. For Λ = Ťm i=1 ti+L, a fundamental domain of Lwill be called an L-fundamental domain of Λ. It then is a fundamental domain of Λ under the action of the translation lattice L. 4.1.1 Euclidean lattices If Rn is equipped with an inner product bwe can associate certain geometric notions to any lattice. To make clear which inner product is used, we will write (L,b) instead of L, whenever b is not clear from context. If we want to emphasize the existence of a inner product we speak of a Euclidean lattice. We write vol for the Lebesgue-volume on the standard Euclidean space and volb for the volume on (Rn,b). By use of the formulae volb = a det(G(b))vol, and vol(FB(L)) = |det(B)| = a det(GB(L,b)) we observe that each fundamental parallelotope has the same volume and we call this number the volume of L: vol(L,b) := volb(FB(L)) = a det(L,b). 4.1.2 Isometries of Euclidean lattices and periodic sets The notion of isometry (cf. 1.1.2 and 1.3.2) can be applied to periodic sets in Euclidean space too. Let Λ = Ťm i=1 ti + L, where w.l.o.g. t1 = 0 and L is on Rn, be a periodic set in (Rn,b). We set GL(Λ) := tU P GLn(Z) | UΛ = Λu, a subgroup of GLn(R), which is conjugate to GLtn(Z) := GL(Λt), and its orthogonal group O(Λ,b) := tU P GL(Λ) | b(Ux,Uy) = b(x,y) @x,y P Λu. It is a well-known and important fact that the orthogonal group of any Euclidean lattice is finite (cf. Lemma 3.1.6) and by O(Λ,b) Ă O(LΛ,b), where LΛ is the maximal translation lattice for Λ, this also is true for periodic sets. 56 4.1. LATTICES AND PERIODIC SETS 4.1.3 Euclidean lattices and quadratic forms We will often prefer to work with the notion of a (positive definite) quadratic form instead of that of an inner product. If Q is any real positive definite quadratic form we obtain an inner product bQ(x,y) = Q(x + y) ´ Q(x) ´ Q(y), which is the associated bilinear form. The other way around, if b is an inner product, we obtain a quadratic form Q by the relation Q(x) = b(x, x) for all x P Rn and b = 12bQ. Thus we will identify the Euclidean space (Rn, 12bQ) with the quadratic space (R n,Q) in any such situation and all of the above stays valid. Since real positive definite quadratic forms are, by choice of basis and their associated Gram matrices, in bijection with the cone of positive definite symmetric matrices Sną0 over R. We will also use the above notation for matrices Q P Sną0. 4.1.4 Parameter spaces for lattices If L is any lattice on (Rn,b) we can use the Cholesky decomposition of the Gram matrix Q P Sną0 of b with respect to the standard basis E (or by slight modification any basis) Q = GE(b) = ATA and any basis matrix B of L to construct a lattice L 1 := (AB)Zn in n-dimensional Euclidean standard space, that is isometric to L. We note that if L = BZn, L 1 = B 1Zn are lattices in Euclidean standard space Rn, they are i. equal, if and only if there exists a U P GLn(Z) such that B 1 = B ¨U; ii. isometric, if and only if there exists a U P On(R) such that B 1 = U ¨ B. Thus we obtain parameter spaces (up to isometry) for lattices on Rn ÐÑ GLn(R)/GLn(Z) isometry classes of lattices on Rn ÐÑ On(R)zGLn(R)/GLn(Z). Similarly the lattice Zn in (Rn,Q 1), with Q 1(x) := Q(Bx), is isometric to L, thus GE(Q 1) = GB(L,Q). Since real positive definite quadratic forms are in bijection to Sną0, the cone of real positive definite symmetric n ˆ n-matrices, we identify a real positive definite quadratic form with its Gram matrix with respect to the standard basis. We note that (Zn,Q) – (Zn,Q 1) if and only if Q,Q 1 are conjugate under the action of GLn(Z). Thus we obtain parameter spaces for bases up to orthogonal transformations of Rn ÐÑ Sną0 isometry classes of lattices on Rn ÐÑ Sną0/GLn(Z). PROPOSITION AND DEFINITION 4.1.1 (COORDINATE REPRESENTATION OF A EUCLIDEAN LATTICE). Let L be a Euclidean lattice on (Rn,Q). We can associate to L and any basis B of L the Euclidean lattice (Zn,GB((L,Q))). This lattice is called the coordinate representation of L with respect to B. Furthermore 57 4. GEOMETRIC FOREPLAY i. (Zn,GB((L,Q))) – (L,Q), ii. the coordinate representations of isometric lattices are conjugate under the action of GLn(Z), iii. coordinate representation induces a bijection between Sną0/GLn(Z) and the set of isometry classes of Euclidean lattices on Rn. 4.1.5 Parameter spaces for periodic sets To extend the above parameter spaces for lattices to periodic sets in general, we will accept some redundancy. We are interested in questions regarding periodic sets only up to isometry. So it is no loss of generality to confine our parameter spaces to comprise of periodic sets for which we assume that t1 = 0. Furthermore we will not parametrize all periodic sets at once, but give one parameter space for eachm P N that comprises only those that are at mostm-periodic. That said, we use the parameter spaces for lattices above, leading to a basis parameter space On(R)zGLn(R)/GLn(Z)ˆ (Rn)m´1 , and a coordinate parameter space Sn,mą0 := S ną0/GLn(Z)ˆ (Rn)m´1 , contained in Sn,m := Sn/GLn(Z)ˆ (Rn)m´1 . While we refer to elements Q of Sn as quadratic forms, we will refer to elements (Q, t) of Sn,m as periodic forms (cf. section 3.2.1 in [Sch09]). We should recall that the above parameter spaces do not give systems of representatives of isometry classes, as was the case for lattices. A periodic set may have several distinct representations of the above forms, since simply replacing t by some t 1 where the t 1i are translates of the ti by some lattice vectors, or are a permutation of the ti gives us a wealth of unequal representations. We say that Λ is representable by Λt, or t-representable if and only if there is a Rn-isomorphism that maps Λ to Λt. If Λ is t-representable, we can associate to any (Λ,Q 1), for Q 1 P Sną0, some isometric (Λt,Q), with Q P Sną0, similar to the case of lattices. A periodic form (Q, t) and a periodic set (Λt,Q) can be thought of being essentially the same object, we therefore also say that (Λ,Q 1) is represented by (Q, t), in the above scenario. Clearly, if Q P Sną0, then (Λt,Q) is a t-representable periodic set, and if U P GLtn, then (Λt,Q) and (Λt,UTQU) are isometric. PROPOSITION AND DEFINITION 4.1.2 (t-COORDINATE REPRESENTATION OF A PERIODIC SET). Let t P (Rn)m´1. Let (Λ,Q 1) be a t-representable Euclidean periodic set on Rn. For any Q P Sną0 for which (Λ,Q 1) is represented by (Q, t), we say that the periodic set (Λt,Q) is a t-coordinate representation of (Λ,Q 1). This construction implies that there is a surjection between Sną0/GL t n(Z) and the set of isometry classes of t-representable periodic sets on Rn. REMARK 4.1.3. The book [Sch09] covers periodic sets and forms with respect to the packing and covering problem. The above presentation relies on the notation used there.  58 4.2. POLYTOPES 4.2 POLYTOPES 4.2.1 Polytopes and polytopal complexes This section follows the treatment in [Zie95]. The dimension of a polytope is the dimension of its affine hull. If a face has dimension k, then we say it is a k-face. For a polytope of dimension nwe, as usual, refer to 0-faces as vertices and n´ 1-faces as facets. A polytopal complex is a set C of polytopes in Rn such that H P C, for P P C every facet of P is in C, and for P,Q P C, P X Q is a face of both P and Q. Its underlying set is |C| := ŤPPC P. If C 1 Ă C is a polytopal complex, we call it a subcomplex of C. If v is a vertex of the polytopal complex C then the star of v in P is star(v,C) := tF P C | v P Fu, the subcomplex containing all faces of C which contain v. The face lattice of a polytope P is the partially ordered set (poset) of all faces of P, ordered by inclusion. It is a lattice in the order-theoretic sense: it is a bounded poset in which every two elements have a unique upper and lower bound (cf. Definition 2.5 in [Zie95]). It is defined analogously for a polytopal complex. Each polytope P gives rise to a complex C(P), the complex of all faces of P. The subcomplex C(BP) formed by all proper faces of P is called the boundary complex of P, and we have |C(BP)| = BP. For a polytopal complex C we fix the abbreviations C(k) for the subset consisting of k-faces of C. We write vert(C) = C(0). A polytopal subdivision of a set M is a polytopal complex C such that |C| =M. A triangulation of a setM is a polytopal subdivision T ofM such that each S P C is a simplex. A triangulation is pyramidal if all of its full dimensional simplices have a point in common, which then is called the apex of the triangulation. We say that C(P) is the trivial subdivision of P. Let T be a pyramidal triangulation of a polytope P, such that the apex 0 of T is in the interior of P, T(0) = vert(P)Yt0u, and TXF is a triangulation without new vertices of F for every F P C(P)(dim(P)´1). A pyramidal triangulation T of a polytope P is essentially identifiable with a triangulation of the bound- ary complex of P, that does not contain points besides the vertices of P. They can be identified by the process of taking the complex of pyramids over 0 in the one direction and removing the faces that do contain 0 in the other. A way to obtain a triangulation without new vertices is by pulling refinements1: let C be a polytopal subdivision and let v P vert(C). Define pv´ (C) by pv´ (C) := ď FĂCPC dim(F)=dim(C)´1 vPC, vRF C(conv(FY tvu)) Y ď CPC vRC C(C), (4.1) that is, pv´ (C) is obtained from C by the following rules: for C P C i. If v R C then C P pv´ (C). ii. If v P C then for all facets F of C: C(conv(FY tvu)) Ă pv´0(C). 1There are other triangulations that would work as well, the so called lexicographic triangulations, cf. 4.3 in [DRS10] 59 4. GEOMETRIC FOREPLAY Discussions of pulling, and more generally, lexicographic triangulations, can be found in both [DRS10] (Definition 4.3.7), and [Lee90]. Our presentation follows the latter source. It seems that the pulling triangulation was first described in [HS69], Lemma 1.4. All of the succeeding can be formulated ac- cordingly for general lexicographic triangulations. LEMMA 4.2.1. Let P be a polytope. i. Let C be a polytopal subdivision of P, and let v P P. Then the above defined pv´ (C) is a refinement of C. ii. Let C(P) be the trivial subdivision of P, and let vert(P) Ă V Ă P such that V is finite and labeled2 in t1, . . . ,ku. Then the subdivision pv´k . . .pv´1(C(P)) is in fact a triangulation of P with vertices in V . iii. If in the above case V = tv1, . . . , vku Y tv0u, where vert(P) = tv1, . . . , vku is labeled in t1, . . . ,ku and 0 labels a point v0 inside P, the triangulation pv´k . . .pv´1pv´0(C(P)) is pyramidal with apex v0. Any such triangulation (resp. subdivision) obtained by a labeling J as above will be referred to as the pulling triangulation (resp. subdivision) with respect to the labeling J. PROOF. (i) and (ii) are dealt with in the sources referred to above. Ad (iii): The first refinement pv´0(C(P)) consists of two classes of polytopes by the above rules. Firstly, all faces of P, since v0 R F for all faces F of P. Secondly, all pyramids of faces of P over v0, since v0 P P, but v0 R F for all facets F of P, which implies that all elements of C(conv(F Y tv0u)) are added for all facets F. Note that only the second step is actually required here. The claim then follows by (i). It is clear from this definition of the pulling triangulation that it can be read off the from face lattice of the polytope; that is, if we have a polytope Pwith k vertices labeled in some ordered set I of cardinality k, we can compute a pulling triangulation of it simply in terms of the image of the face lattice of P in 2I: all the necessary information, which is of purely combinatorial nature, is accessible from there. In this sense we allow to speak of the pulling triangulation of an abstract lattice F, that is (isomorphic to) the face lattice of some polytope: Given a lattice F P 2I, at the i-th refinement step (for i P I) we run through all F P F and check whether i R F to keep F part of the refinement or else replace F by the collection of sets F 1 Y tiu for each maximal subset F 1 Ă Fztiu. Again this can easily be extended to arbitrary lexicographic triangulations. LEMMA 4.2.2. Let S be any set of combinatorially equivalent polytopal complexes with k vertices and let I be an ordered set of cardinality k. Let F be a representative of their face lattices in 2I. For each C P S fix an isomorphism from F to the face lattice FP of P. Then 2By a labeling by some ordered set J we mean to express that there is some bijective map from J to V . This is depicted by enumerating the vertices in the above. 60 4.3. POINT SETS AND TILINGS IN EUCLIDEAN SPACE i. I is a labeling for the vertices of P, ii. for every abstract pulling (lexicographic) triangulation T of F, the image TP of T under the extension of φP defined on T is a pulling triangulation of P. This Lemma includes the case (iii) of Lemma 4.2.1. This construction will be applied later on (cf. Proposition 6.2.1). REMARK 4.2.3. A little note on the side: If P and P 1 are combinatorially equivalent polytopes, for which we fix a labeling of the vertices, in general it will not be true that a triangulation of P, given in terms of the labels, also gives a triangulation of P 1. An example can be given by two octahedra: A C B D E F A C B D E F In the right octahedron the complex generated by the 3-simplices convtA,B,C,Du, convtA,C,D,Eu, convtA,B,C,Eu, convtA,B,D, Fu, and convtB,C,D, Fu is a triangulation. The translation of this triangulation in terms of the labels to the left octa- hedron fails to be a subdivision, in particular, it is not a triangulation: in the right octahedron convtA,B,C,Du (the red one) is a simplex, it is not full dimensional in the left octahedron. Even if we would omit this polytope we would not obtain a subdivision of the left octahedron, the intersection of 3-simplices on different sides of convtA,B,C,Du is never a common face of both. However, if P and P 1 are equivalent in a stronger sense, namely if they have the same oriented matroid, then all triangulations can be translated (cf. Corollary 4.1.44. in [DRS10]). In our applica- tions later on we will not be lucky enough to consider families of polytopes that are equivalent in this stronger sense. The mentioned defect of the polytope convtA,B,C,Du not being of the same dimension in both octahedra is actually a proof of the inequivalence their oriented matroids. But as stated above: not all hope is lost. If P and P 1 are combinatorially equivalent, then at least the lexicographic triangulations can be translated and we can even do so for polytopes with an interior point, used in a pyramidal triangulation.  4.3 POINT SETS AND TILINGS IN EUCLIDEAN SPACE In the subsequent (V ,b) is an n-dimensional Euclidean space with norm }x} := ab(x, x) and affine isometry group Iso(V ,b) = tφ P AGL(V) | @x,y P V : }φ(x)´ φ(y)} = }x´ y}u. 61 4. GEOMETRIC FOREPLAY 4.3.1 Tilings A polytopal tiling of V is a set T of full dimensional polytopes, such that Ť PPT P = V , where for arbitrary P,P 1 P T the intersection PXP 1 is a polytope of at most dimension n´1. Thus it is a covering and packing of space by polytopes. A tiling is face-to-face if for arbitrary P,P 1 P T the intersection P X P 1 is a face (possibly empty) of both P and P 1, that is, if and only if T is the set of full dimensional polytopes of a polyhedral complex. A tiling is facet-to-facet if for P,P 1 P T with n ´ 1-dimensional intersection PXP 1, this intersection is a facet of both P and P 1. A tiling is locally finite if each bounded set in V intersects only finitely many elements of T. If T is a locally finite polytopal tiling, it is face-to-face if and only if it is facet-to-facet (cf. [GR89]). In this case T induces a subdivision of the embedding space. A tiling T for which all tiles are congruent to one another is called it is called monohedral. An isometry of a tiling T in a Euclidean space V is an affine isometry φ of V that fixes T under the induced action, that is for all P P T also φ(P) P T. If the isometry group of a tiling T operates transitively, the tiling is called isohedral (cf. [Eng93]). 4.3.2 Delone sets A set Λ Ă V is called discrete if and only if for each v P Λ there is some } ¨ }-ball Bε(v) Ă V such that Bε(v) X Λ = tvu. A discrete set P Ă V is a Delone set or (r,R)-system if Br(p) X P = tpu for all p P P and if BR(x)XP ‰ H for all x P V . Thus P is a Delone set if and only if the distance of arbitrary points p,p 1 P P is at least r and from each point x P V there is a point p P P of distance at most R. A vertex set having these properties is referred to as being uniformly discrete and relatively dense respectively. A Delone set P is said to be symmetric if the (affine) isometry group Iso(P,b) := tφ P Iso(V ,b) | φ(P) = Pu, of P acts transitive on P. An important example of Delone sets are Euclidean lattices and periodic sets. Furthermore lattices are always symmetric Delone sets. 4.3.3 Dirichlet-Voronoi cells and tilings of Delone sets We will now meet the main object of interest of this Part: DEFINITION 4.3.1 (DIRICHLET-VORONOI CELL). Let (V ,b) be a Euclidean space with norm } ¨ }, and let P Ă V be discrete. The Dirichlet-Voronoi cell (or DV-cell) of an element p P P relative to P and } ¨ } is the set DVP,}¨}(p) := x P V }x´ p} ď }x´ q}, @q P P ( . ˛ Since the Dirichlet-Voronoi cell of a point of a discrete set P is defined as an intersection of half-spaces, it is natural to wonder if a finite number of half-spaces suffice and thus if it is a polyhedron in fact. Now P being discrete does not suffice to ensure this: there exist infinite discrete vertex sets such that 62 4.3. POINT SETS AND TILINGS IN EUCLIDEAN SPACE there are non-polyhedral cells (cf., Example 2.1 in [Voi08a], also available as Beispiel 3.2.1 in [Voi08b]). However if P is finite (obviously), or a Delone set this can not happen, in such a case every Dirichlet- Voronoi cell is a polytope (cf. Theorem 2.2 in [Eng93] for the latter). In any case, the Dirichlet-Voronoi cells of discrete P will be generalized polyhedra, that is every intersection of such a cell with a poly- tope is itself a polytope (cf. Proposition 32.1 in [Gru07]). Even though all Dirichlet-Voronoi cells we will deal with will be polytopes, we prefer the name "cell". At a later point (Definition 4.3.13 below) we will define another object, closely related to the notion of Dirichlet-Voronoi cell. This new object will be called a Dirichlet-Voronoi polytope, so some care in handling these objects is in order. Of direct interest to us is the collection of all Dirichlet-Voronoi cells of points in P with respect to the norm } ¨ }: DVP(} ¨ }) := DV(p) p P P ( . PROPOSITION 4.3.2. Let P be a Delone set in Euclidean space (V ,b) with norm } ¨ }. Then DVP(} ¨ }) is the set of full dimensional cells of a polytopal subdivision of V . The statement of the Proposition is well-known, cf., Corollary 32.1 in [Gru07] (being facet-to-facet) and Theorem 2.2 in [Eng93] (being polytopal). The polytopal subdivision generated by DVP(}¨}) is called the Dirichlet-Voronoi subdivision of (V ,b) relative to P. If P is a Delone set, such that DVP(} ¨ }) is monohedral (isohedral), we will by abuse of language call P itself monohedral (isohedral). REMARK 4.3.3. Note that the notion of a Dirichlet-Voronoi cell and tiling still make sense if we use a positive semi-definite bilinear or quadratic space. However, Dirichlet-Voronoi cells might be unbounded polyhedrons.  PROPOSITION 4.3.4. Let P be a discrete set. If φ is an affine isometry of P, then DVP,Q(φ(x)) = φ (DVP,Q(x)) . PROOF. By definition: v P DVP,Q(φ(x))ô }φ(x)´ v}Q ď }p´ v}Q @p P P. Since φ is an isometry, this is equivalent to }x ´ φ´1(v)}Q ď }φ´1(p) ´ φ´1(v)}Q @p P P which by φ(P) = P can be expressed as }x ´ φ´1(v)}Q ď }p ´ φ´1(v)}Q @p P P. The latter clearly is equivalent to φ´1(v) P DVP,Q(x), as was claimed. COROLLARY 4.3.5. Let P Ă V be a symmetric Delone set. Then DVP(} ¨ }) is an isohedral polytopal tiling of V . In particular, a symmetric periodic set is a monohedral periodic set. PROOF. If p,p 1 P P, there is an isometry of P mapping p to p 1 since P is symmetric. The Proposition above showed that any such isometry gives rise to an isometry of DVP(Q), mapping DVP,Q(p) to DVP,Q(p 1). Thus already the set of these isometries of DVP(Q) acts transitive. The above Corollary holds in particular in the case of a lattice (L,Q), where already the translations by lattice elements operate transitive on the set of all Dirichlet-Voronoi cells and DVL,Q(x) = x+DVL,Q(0). 63 4. GEOMETRIC FOREPLAY We will usually abbreviate DV(L,Q) = DVL,Q(0), as is common in the literature on lattices. If we deal with a coordinate representation (Zn,Q) of a lattice, we might further abbreviate DV(Q) = DV(Zn,Q); similarly we write DV(L) = DV(L, x, y) = DV(L, In) if L is a basis representation in Euclidean standard space. More generally, assume that P is a symmetric Delone set in (V ,Q) and that 0 P P. We will write DV(P,Q) := DVP(0) and refer to it as the Dirichlet-Voronoi cell P, which seems justifiable by virtue of the transitive operation of Iso(P,Q). If Q = In, i.e., P is in Euclidean standard space, we abbreviate further and write DV(P). In the case P = Λt on (V ,Q), we write DV(Q, t) in resemblance of the above definition for lattices. We give another example of symmetric Delone sets. LEMMA 4.3.6. Let L be a lattice and v R L. Then Λ := L Y˙ v + L is a symmetric periodic set. In fact, if T(L) denotes the group of translations by elements of L and φ 1 2v denotes the point reflection with center 12v, then T(L)¸ xφ 1 2v y ă Iso(Λ) acts transitively on Λ. PROOF. Let τx denote the translation by x. Then φ 1 2v is given by ´ id ˝ τ´v = τ 1 2v ˝ ´ id ˝ τ´ 12v, it is an isometry, and interchanges L and v+ L:# ´ id ˝ τ´v(x) P v+ L for x P L, ´ id ˝ τ´v(x) P v for x P v+ L. Since T(G) acts transitively on L the claim follows immediately. For monohedral periodic sets, in particular for lattices, the volume of a Dirichlet-Voronoi cell and the determinant of a translation lattice are closely related. LEMMA 4.3.7. Let Λ be a monohedral periodic set in (Rn,b), which is m-periodic for a translation lattice L. Then volb(DV(Λ)) = vol(L,b) m = a det(L,b) m . PROOF. The lattice case first: DV(L) is a fundamental domain of L (cf. 4.3.2), therefore volb(DV(L)) = volb(FB(L)) = vol(L,b). If Λ is as above, then F := Ťm i=1 DVΛ(ti) is a fundamental domain for L. Thus volb(DV(Λ)) = 1 m volb(F) = 1 m volb(FB(L)) = vol(L,b) m . So in the case of monohedral periodic sets it makes sense to speak of the volume of such a set if we understand that this is the volume of its Dirichlet-Voronoi cell. 4.3.4 Delone polytopes of point sets Though we are ultimately interested in Dirichlet-Voronoi cells, Delone polytopes and subdivisions will play a crucial role. 64 4.3. POINT SETS AND TILINGS IN EUCLIDEAN SPACE DEFINITION 4.3.8 (DELONE POLYTOPE). Let P be a discrete vertex set and let P Ă V be a polyhe- dron with vertex set vert(P) Ă P. P is a Delone polyhedron (with respect to P and } ¨ }) if there exists c P V and r P R with }c´ p} ě r for all p P P, where }c´ p} = r if and only if p P vert(P). ˛ The definition above is equivalent to P satisfying the empty-sphere property: P is a Delone polyhedron if and only if there exists a } ¨ }-ball B = Br(c) Ă V such that BX P = vert(P). In general, an arbitrary discrete set in a Euclidean space does not have to admit Delone polyhedra. However if P Ă V is a Delone set, they exist and are polytopes due to the definiteness of b and discreteness of P. Even better: they make up a polytopal subdivision of the enveloping space. This can be seen as follows: for any sphere S in (V ,b) that contains no point of P we construct the polytope P := conv(S X P). Then P is a Delone polytope, we say that it is constructed by the empty-sphere method. It is full-dimensional if the intersection of P and S contains an affine basis. PROPOSITION 4.3.9. Let P be a Delone set in (V ,b) with norm } ¨ }. Then the set of polytopes constructed by the empty-sphere method is the polyhedral complex generated by the full-dimensional Delone polytopes. PROOF. This is the combination of Proposition 32.2 and Theorem 32.1 in [Gru07], there stated as Corol- lary 32.1 for Delone triangulations, though valid for arbitrary Delone subdivisions. DEFINITION 4.3.10. Let P be a Delone set in (V ,b). The polytopal complex of Proposition 4.3.9 is called the Delone subdivision of (V ,b) relative to P. We write DelP(} ¨ }) for this subdivision. ˛ There is an alternative way of constructing the Delone subdivision of a Delone set, by a lifting construc- tion, which we mention very briefly. Let P Ă V be a Delone set and let ω : V Ñ V ˆ R; x ÞÑ (x, }x}2). Then conv(ω(P)) is a generalized polyhedron. All faces of this generalized polyhedron are lower in the following sense: if F is a face of P and x P F, then for all ε ą 0 the point x + ε(0,´1) P V ˆ R is not contained in P. Now one can check that the projection of the (lower) faces of P back to V (i.e., forgetting the last coordinate) gives a polytopal complex, which is in fact the Delone subdivision of V (cf. 32.1 in [Gru07]). We are mostly interested in the case where P is a periodic vertex set embedded in a quadratic space (V ,Q). In this case we write DelP(Q) for DelP(} ¨ }Q). REMARK 4.3.11. Note that the above Definitions and Proposition also make sense if we consider a positive-definite bilinear space. What changes is that now there can be unbounded Delone polyhedra. But they are still constructible by the empty-sphere method and make up a polyhedral complex, which we will then also refer to as Delone subdivision (cf. (1.7) in [Nam76]) or 2.1 in [Val03]).  REMARK 4.3.12. It is also possible to define Delone and Dirichlet-Voronoi subdivisions by a lift- ing construction, we refer the interested reader to Chapter 32 in [Gru07].  4.3.5 Dirichlet-Voronoi polytopes in dual space We will now define what we understand to be a Dirichlet-Voronoi polytope. These objects are closely related to Dirichlet-Voronoi cells, but have certain advantages for us, in particular when dealing with coordinate representations of Euclidean lattices L = (Zn,Q). This definition is, in its generality, due to [Nam76] in the case of P = Zn, but all of the properties we are interested in here also hold for the 65 4. GEOMETRIC FOREPLAY case of an arbitrary Delone set. Namikawas definition resembles Voronoi’s original approach to what is now called a Dirichlet-Voronoi cell (cf. Equation (10) in [Vor08]). The usefulness of this Definition in a similar context can already be seen in [Val03], where the main application is the lattice covering problem. In particular we adapt the modern notation used by Vallentin. DEFINITION 4.3.13 (DIRICHLET-VORONOI POLYTOPE). Let (V ,b) be a Euclidean space with norm } ¨ } and quadratic form Q P Sną0 with associated bilinear form b = bQ. Let P Ă V be a Delone set and P = convtp1,p2, . . .u be a Delone polyhedron of Q in P. The polytope DV(Q,P) := bˆ(x) P V˚ }x´ pi} ď }x´ p}, @p P P ( is called the Dirichlet-Voronoi polytope of Q corresponding to P with respect to P. ˛ The object defined above actually is a bounded polyhedron, thus a polytope, even if we apply the above definition to the case of a positive-semidefinite quadratic form (cf. (1.7) in [Nam76]). Recall that the map bˆ : V Ñ V˚ is given by bˆ(x) = b(x, ¨) (cf 1.1.5). Now if Q P Sną0, it is the quadratic form associated to the Euclidean space (Rn, 12bQ). Since bQ and 1 2bQ are just scaled versions of one another, it actually does not make a difference which of the two bilinear forms we use for the definition. In particular DV(Q,P) = DV(λQ,P) for all λ P Rą0. The concept of Dirichlet-Voronoi cell and polytope are closely related, we state it as a Lemma. LEMMA 4.3.14. Let P be a Delone set and Q P Sną0 with associated bilinear form b = bQ. Then for p P P DVP(Q, tpu) = bˆ(DVP,Q(p)). In particular DVP,Q(p) and DVP(Q, tpu) are affinely isomorphic. In particular for the case of a lattice (L,Q) we obtain that DVL(Q, t0u) = bˆ(DV(L,Q)). Analogously for anm-periodic set Λ with translation vectors t1, . . . , tm DVΛ(Q, ttiu) = bˆ(DVΛ,Q(ti)). 4.3.6 Duality of Dirichlet-Voronoi and Delone subdivisions Dirichlet-Voronoi and Delone subdivisions are combinatorially dual objects. This becomes even more apparent if we switch Dirichlet-Voronoi cells with the associated Dirichlet-Voronoi polytopes. The following properties of Delone and Dirichlet-Voronoi polytopes will be useful, they can be found as part of (1.4) in [Nam76] if we replace Zn by P. PROPOSITION 4.3.15. Let Q P Sną0, P be a Delone set. i. For P,P 1 P DelP(Q) we have that P is a face of P 1 if and only if DV(Q,P 1) is a face of DV(Q,P). ii. For P P DelP(Q) we have dim(P) + dim(DV(Q,P)) = n. 66 4.3. POINT SETS AND TILINGS IN EUCLIDEAN SPACE By identifying Rn with Rn˚ by means of the standard inner product, it is easy to derive the following Lemma. LEMMA 4.3.16. Let P be a Delone set. For Q,Q 1 P Sną0 with DelP(Q) = DelP(Q 1), the normal fans of the Dirichlet-Voronoi polytopes DV(Q,P) and DV(Q 1,P) coincide. SKETCH OF PROOF. Represent DV(Q,P) by elements of the form Qx of Rn. Then all (not necessarily relevant) defining inequalities of this polytope look like 2(pi ´ p)TQx ě }pi}2Q ´ }p}2Q, for suitable p P P and pi P vert(P). In particular this holds for those inequalities relevant to a given face. In any case this implies that forQ,Q 1 as assumed, the polytopes are described by a systemAx ě b andAx ě b 1 respectively (where the i-th row ofA is a suitable 2(pi´p)T ), but this implies the claim. This in particular implies that the facet normals are characterized completely by ∆P(Q) and coincide for quadratic forms that induce the same Delone subdivision. The duality of the Delone and Dirichlet-Voronoi subdivision of a space (V ,Q) implies a straightfor- ward characterization of the vertices of the Dirichlet-Voronoi cell of a point by its star in the Delone subdivision: LEMMA 4.3.17. Let P be a Delone set in a quadratic space (V ,Q) with norm } ¨ }Q. Then vert(DVP,Q(p)) = tcentroidQ(D) | D P star(p,D), dim(D) = nu, where centroidQ(D) is the center of the } ¨ }Q-circumsphere of D. 67 CHAPTER 5 Periodic vector quantization This chapter contains a brief presentation of some of the main notions regarding vector quantization. It is included for the sole purpose of providing a self-contained overview about the main notions of vector quantization for a reader not yet familiar with this subject. We try to make a case for the importance of the quantizer constant within this problem. For a more detailed accounts we refer to [GG12] and [Zam14], the subsequent discussion is strongly based on these sources. 5.1 THE NORMALIZED SECOND MOMENT AND THE QUANTIZER CONSTANT Given a body1 P Ă Rn and a point pxwe define the second moment of P about px by U(P) := ż P }x´ px}2dx, the normalized second moment I(P) := U(P) vol(P) , and finally the dimensionless normalized second moment G(P) := 1 n U(P) vol(P)1+2/n = 1 n I(P) vol(P)2/n . We use the notation that is more common in the mathematical literature, as used by Conway and Sloane, cf. [CS98, Chapter 21]. However one should note that in much of the literature in information or coding theory, the name second moment is used for what we call the dimensionless second mo- ment, denoted σ2(P) = 1 n U(P), and the term normalized second moment is as well understood to be dimensionless, cf. [Zam14]. We now use the notation of Chapter 6 concerning Dirichlet-Voronoi cells. For a lattice Lwe set G(L) := G(DV(L)) 1We do not get into technicalities here: we only need this notion for convex polytopes and ellipsoids in the remainder. For such objects the notions discussed here clearly make sense. 69 5. PERIODIC VECTOR QUANTIZATION and for any periodic set Λ with congruent Dirichlet-Voronoi cells we also abbreviate G(Λ) := G(DV(Λ)). The use of lattices and periodic sets in quantization motivates another name for G(Λ), it is sometimes referred to as the quantizer constant of Λ. We will usually use this nomenclature from here on. Thus for a lattice L in Euclidean standard space G(L) = 1 n 1 det(L)1/2+1/n ż DV(L) }x}2dx, and for anm-periodic set Λ = Ťm i=1 ti+L in Euclidean standard space, for which all Dirichlet-Voronoi cells are congruent G(Λ) = 1 n m1+2/n det(L)1/2+1/n ż DV(Λ) }x}2dx, by relating the volume of a Dirichlet-Voronoi cell to the determinant of the lattice involved (cf. Chapter 6, Lemma 4.3.7). If we are in the more general situation of a Euclidean space (R,b), whereQ is the Gram matrix of some inner product b, we find for a lattice Gb(L) = 1 n 1 (det(Q)1/2 det(L))1/2+1/n ż DV(L,Q) }x}2Qdx, and for anm-periodic set Λ = Ťm i=1 ti+L in Euclidean standard space, for which all Dirichlet-Voronoi cells are congruent Gb(Λ) = 1 n m1+2/n (det(Q)1/2 det(L))1/2+1/n ż DV(Λ,Q) }x}2Qdx. In this situation we also think of Gb(Λ) as "a quantizer constant", but now referring to a different distortion measure. This will be elaborated on in 5.2.2 and 5.2.4 below. 5.2 VECTOR QUANTIZATION 5.2.1 General notions for vector quantization In information and coding theory a (finite) vector quantizer of dimension or block length n and size N is given by specification of a finite set C of cardinality N and a map Q : Rn Ñ C, where C is called the codebook, its elements the codewords or code vectors, and Q is known as the quantization rule. We assume that the elements of C are indexed by t1, . . . ,Nu. An associated quantity is the resolution r := (log2N)/n, also called code rate. To each c P C there is associated a quantizer cell Q´1(c) = tx P Rn | Q(x) = cu, and the collection of all cells gives a partition of Rn. We divide cells into two types, either they are bounded and called granular cell, or they are unbounded and called overload cell. The collection of either of those is then refered to granular and overload region respectively. A vector quantizer is called regular if all of its cells are convex, and it is called polytopal if each cell is a polyhedron. 70 5.2. VECTOR QUANTIZATION To describe and analyze the application of vector quantizers in source coding it is usually decomposed into two seperate objects. On the one hand we have the encoder E : Rn Ñ t1, . . . ,Nu and on the other hand the decoderD : t1, . . . ,Nu Ñ Rn. We do not wish to go into detail here and refer to [GG12, section 10.1]. The idea of encoding is that the encoder first chooses a suitable c P C by finding the cell that contains a given source vector and then determines the index of said cell. Both tasks can become arbitrarily difficult, but can be dealt with for certain structured vector quantizers as we will elaborate on in a bit. 5.2.2 The performance of a vector quantizer Let X be continuously distributed random vector in Rn with probability density function fX. Let d be a distortion measure, that is, a nonnegative function of X and its reproduction Q(X) under quantization. We assume that fX is a joint probability density function for the components of X. Let tXtu be a (discrete time) random process constituted of random vectors, we assume from here on that it is independent and identically distributed (abbreviated by i.i.d.). If this process is stationary and ergodic we find, that with probability one the limit on the left side exists and satisfies lim nÑ∞ 1n nÿ t=1 d(X,Q(X)) = E(d(X,Q(X))). Here limnÑ∞ 1n řnt=1 d(X,Q(X)) is the time average of the distortion and D = E(d(X,Q(X))) is the average distortion of the coding scheme (cf. 10.3 in [GG12]). For the vector quantization scheme this can then be expressed by D = ż d(x,Q(x))fX(x)dx (5.1) or with the partition of Rn given by the quantizer cells as D = ÿ cPC PcE[d(x, c) | X P Q´1(c)] where for c P C we set Pc = P(X P Q´1(c)) to be the probability of X lying in the cell associated to c. Of particular interest is the squared error distortion measure given by the squared Euclidean distance of x and its reproduction Q(x) d(x,Q(x)) = }x´Q(x)}2. The reason behind this choice lies rather in the mathematical tractability of this measure, than its actual performance for various media. In any case, it can be interpreted as the energy of power of an error, which gives it some physical significance (cf. 2.4 in [Gra12]). We also want to introduce the weighted squared error measure given by some positive definite bilin- ear form b through d(x,Q(x)) = b(x´y, x´y) = }x´y}2b, where the Mahalanobis distortion measure is a special case, with b chosen in such a way that the Gram matrix of b with respect to the standard basis is given by the covariance matrix of the source vector X. In the case of an i.i.d. Gaussian source with unit variance and zero mean the Mahalanobis distortion measure and the squared error coincide. The squared error distortion measure also falls into the realm of single letter distortion measures of the form d(X, Xˆ) = řn i=1 dm(Xi, Xˆi), where dm is a scalar distortion measure, the per-letter distortion measure. 71 5. PERIODIC VECTOR QUANTIZATION Two well-known optimality criteria in vector quantization are the nearest neighbor condition and the centroid condition. The nearest neighbor condition states that given a codebook C and a distortion measure d the optimal choice of the quantizer rule, and thus cell decomposition of Rn associated to Q, has to satisfy Q(x) = cñ d(x, c) = mintd(x, c 1) | c 1 P Cu. It is not an “if and only if” statement because the codeword cminimizing the right hand side might not be unique. Therefore the quantizer cells have to lie between the interior and closure of the Dirichlet- Voronoi cells of the codewords with respect to the chosen distortion measure. If on the other hand the cell decomposition tC1, . . . ,CNu of Rn is given and the codebook has to be chosen, the centroid condition states that the optimal choice of C = tc1, . . . , cNu has to satisfy ci = centroidd(Ci) for all i P t1, . . . ,Nu, where centroidd(C) = arg mintE[d(X, c) | X P C]u. The easy proofs may for example be found in [GG12, 11.2]. 5.2.3 Periodic vector quantization with the squared error distortion measure We now turn our attention to quantizers whose codebooks come from a (monohedral) periodic set Λ Ă Rn - which is assumed to generate Rn. Before focusing on how to quantize, we want to quickly point out that there are two ways to cope with the infinite size of Λ. One approach is to chose a finite subset C, the other idea is to work with the, infinite, periodic set as codebook, but applying some sort of entropy coding to obtain a notion of coding-rate that is finite. The book [Zam14] offers a quite recent and rather detailed introduction and discussion of these information theoretic aspects of vector quantization. Let us only briefly mention, without discussion of the results, that the below rough estimation of the quality of a quantizer in terms of the quantizer constant is completely justified by taking a closer look on the results of information theory: it is always the quantizer constant which has to be as small as possible for a quantization scheme to work well. We concentrate our attention to distortion measures coming from Euclidean geometry, that is in most cases the squared Euclidean distance, but we should also keep the weighted squared error measure in mind. However, as discussed above, it is no restriction of generality to assume that d is the standard squared error distortion measure. There is a widely known and widely accepted conjecture of Gersho regarding vector quantization: the optimal quantizer will have cells that are all congruent to some polytope. This is discussed in his 1979 work [Ger79]. We will therefore restrict our attention to monohedral periodic sets, so that all Dirichlet- Voronoi cells are congruent. We will call the resulting quantizer a monohedral periodic quantizer. Mathematically rather well tractable special cases of monohedral periodic sets are symmetric periodic and, above all, lattices. If Λ is in fact a lattice, we will refer to the resulting quantizer as lattice quantizer. The basic idea is similar in both cases. Let C Ă Λ be the codebook chosen from Λ. Respecting the optimality criteria presented above we start with the Dirichlet-Voronoi subdivision (cf. Definition 4.3.3) of Rn induced by C, and map an element x P Rn to the centroid of the Dirichlet-Voronoi cell it is contained in, or to any one of those centroids of such cells if x is contained in more than one, where a tie may be broken in an arbitrary way. 72 5.2. VECTOR QUANTIZATION The easiest example, which indeed is a lattice quantizer, that albeit captures the mathematical essence of this idea, is that of rounding, which we can interpret as using the lattice Z Ă R with Q(x) = [x]. Thus the quantizer cells are given by the half-open intervals of the form [z´ 1/2, z+ 1/2) for z P Z. Now given a symmetric periodic set Λ Ă Rn and the squared error distortion measure d(x,y) = }x´ y}2 the above summarizes in taking the quantization rule to be given by Q : Rn Ñ Λ; x ÞÑ arg min λPΛ }x´ λ} where ties might be broken in an arbitrary way. The quantizer cell of a given λ P Λ is then essentially given by its Dirichlet-Voronoi cell DVΛ(λ) = tx P Rn | }x´ λ} ď }x´ λ 1}, for all λ 1 P Λu. This is a bit imprecise however, as noted before. To obtain a well-defined mapQwe were in need to break ties, that is a point x P Rn lying on the boundary of DVΛ(λ) may not satisfy Q(x) = λ. This fortunately is not a problem in the analysis or design of such quantizers since DVΛ(λ)o Ă Q(λ)´1 Ă DVΛ(λ), implying that DVΛ(λ)zQ(λ)´1 is a null set, regardless of the way a tie is broken. For the sake of analysis we will therefore assume that equality on the right hand side holds, and by abuse of language we will therefore sometimes refer to DVΛ(λ) as the quantizer cell of λ. Assuming squared euclidean distance we now focus on the average distortion, which we will refer to as mean squared error. The easiest case would be that of a uniform source of finite support S, where C = ΛX S. Then the integral in (5.1) becomes D = ż S }x´Q(x)}2dx = ż SG }x´Q(x)}2dx+ ż SO }x´Q(x)}2dx where SG and SO shall denote the intersection of the granular and overload region with S respectively. For more complicated and possibly unbounded sources lattice quantization is often used under the assumption of high, but finite, resolution. We will formulate the analysis for periodic quantizers and then return to the special case of lattices in a moment. To keep this discussion brief we restrict ourselves to say that under the high resolution assumption we assume that the rate grows so large that the source distribution among each cell becomes approximately uniform (cf. [GG12, 10.6]). We therefore can approximate the average distortion D « ÿ λPC f(λ) ż DVΛ(λ) }x´ λ}2dx, where we assume that fX(v) « f(λ) for all v P DVΛ(λ). With Pλ := Pr(X P DVΛ(λ)) « f(λ)¨vol(DVΛ(λ)) we get D « ÿ λPC Pλ vol(DVΛ(λ)) ż DVΛ(λ) }x´ λ}2dx. This analysis refines when Λ is a monohedral periodic set, since then all Dirichlet-Voronoi cells in the granular region are congruent (cf. 4.3.3)). Therefore in such a case we can approximate the granular contribution to the average distortion by DG « 1vol(DV(Λ)) ż DV(Λ) }x}2dx (5.2) 73 5. PERIODIC VECTOR QUANTIZATION where we assume that almost all inputs fall into the granular region. Under the high resolution as- sumption one usually assumes furthermore that DO is neglectable compared to DG, so that (5.2) be- comes a good approximation of the average distortion itself. If we follow these lines we can express the distortion through the dimensionless normalized second moment of DV(Λ) by D « N ¨ I(DV(Λ)) = N ¨ n ¨ vol(DV(Λ))1+2/n ¨G(DV(Λ)). For a lattice L this simplifies to D « N ¨ I(DV(L)) = N ¨ n ¨ det(L)1/2+1/n ¨G(L), since vol(DV(L)) = a det(L) in this case. It is this relation (and in more sophisticated schemes certain similar relations, cf. [Zam14] for a recent textbook treatment) that justify to call G(Λ) quantizer constant. 5.2.4 Periodic vector quantization with the weighted squared error case distortion measure Surely, if b is some positive definite bilinear form and we consider the euclidean space (Rn,b), and ifQ is the gram matrix of bwith respect to the standard basis, there is an isometry of this space to Euclidean standard space by mapping the standard basis in Rn to any basis that consist of the columns of any matrix A that satisfies Q = ATA. Such a matrix always exists as is well-known, i.e., per Cholesky decomposition. This can be utilized to reduce the more general case of weighted squared error to the case of standard squared error. Clearly if one follows through the above arguments for the squared error distortion measure one obtains similar formulae for the distortion with Gb replacing G. But Gb(Λ) = det(A)´1G(AΛ), as is seen by invoking the formula for change of variables in integration (cf. the reasoning in Lemma 6.3.2). If b is fixed, then this shows that finding the minimal value of Gb is equivalent to finding the minimal value for G. 74 CHAPTER 6 Dirichlet-Voronoi cells and their normalised second moments This chapter contains some of the main results of this thesis. We aim at contributing to certain math- ematical facets of the quantizer problem, given that the codebook is a periodic set. In particular, the most prominent case of lattice codebooks lies at the very center of the subsequent investigations. We derive polynomial optimization problems related to the quantizer problem, a piecewise explicit ex- pression for the quantizer constant of suitable periodic sets, and prove thatA#4 andD # 4 are local minima in the class of 4-dimensional lattices. We will recall some well-known notation and results of Voronoi’s second reduction theory, which we use as a main tool in our investigations. We then apply this theory to start a systematic investigation of the quantizer constant for (monohedral) periodic sets. This generalizes the special result that the root lattice A#3 – D#3 is the sole local and thus global minimum of the lattice quantizer problem in dimension 3 (cf. [BS83]). We fix a rational standard periodic set Λt and take a look at all periodic forms (Q, t), or equivalently all periodic sets (Λt,Q). We can utilize Voronoi’s second reduction theory to obtain a finite partition of Sną0ˆttu/GLtn(Z). This leads to a finite number of local quantizer problems in each dimension for the chosen standard periodic set. Unfortunately, even for lattices, the number of distinct local quantizer problems grows to be inaccessible beyond dimension 5. From this we derive the main results of this part: the general problem of finding the optimal quantizer, with respect to a periodic codebook satisfying certain conditions (cf. Assumption 6.3.1), can be split into a finite number of polynomial optimization problems (cf. 6.3.7). A particular class of periodic sets that satisfy Assumption 6.3.1 without further ado, is the class of lattices. We furthermore show that for the partition of Sną0 into secondary cones, there exists a piecewise explicit expression of the quantizer constant, viewed as a function Sną0 Ñ Rą0. The restriction of this function to each secondary cone can be written as the quotient of a polynomial in Q, depending only on the secondary cone, and a power of det(Q) (cf. 6.3.9). For the three inequivalent secondary cones in dimension 4 we compute these explicit expressions (cf. 6.4.3, in particular, (6.10), (6.11), and (6.12)). We conclude this part with an application of the result to local optimality of lattices in dimension 4 (cf. 6.4.4). 75 6. DV-CELLS AND NORMALISED SECOND MOMENTS 6.1 VORONOI’S SECOND REDUCTION THEORY We sketch out the basic notions of this theory, following the treatment in [Sch09] and [Val03], to keep this thesis as self contained as possible. From now on we will restrict to discrete vertex sets that are periodic, including the classical case of lattices. 6.1.1 Secondary cones of Delone triangulations and subdivisions Let Λ = Λt be a standard discrete periodic vertex set Λt = Ťm i=1 ti + Zn and T be a linear subspace of Sn. We ultimately are interested in classifying all essentially different Delone triangulations ofΛ that occur by varying the quadratic form defining the geometry in Sną0 X T . From a technical point of view it is more appropriate to work in the larger set S˜ně0 X T = conetxxT | x P Znu X T , which is the rational closure of Sną0 X T (cf. Proposition 4.2 in [Sch09]). Suppose Q P Sną0. Then we can construct Delone polytopes for Λ as follows: take n + 1 affinely independent elements of Λ, construct the circumsphere of the simplex they define. If there are points of Λ inside the associated ball, there is no Delone polytope which contains the chosen elements as vertices; if there are no points of Λ inside this ball, the convex hull of the set of all elements of Λ, which lie on the boundary of the constructed ball, is a Delone polytope by construction. This is the empty-sphere method (cf. Section 4.3.4). This does not work for an arbitrary Q P Sně0. However, by the reasoning above it can bee seen to hold for such Q P Sě0 for which there exists a U P GLn Z) such that UTQU = ( 0 0 0 Q 1 ) , with Q 1 P Sną0. In this case the Delone subdivision will contain unbounded polyhedra. As it turns out this is it, a form Q P Sně0 admits Delone polyhedra with respect to Λ if and only if Q P S˜ně0 (cf. [Sch09, 4.1, 4.2], [Nam76, §2.1]). Let D be any Delone subdivision of Λ, then we define the T -secondary cone of D with respect to Λ to be ∆T (D) = Q P S˜ně0 X T DelΛ(Q) = D ( . In the classical case T = Sn we simply speak of secondary cones. Delone triangulations are of primary interest, since the broader class of general Delone subdivisions is closely related to them, by the observation that if D is a Delone subdivision, which is not already a triangulation, there is a Delone triangulation D 1, such that ∆T (D) is a face of ∆T (D 1) (cf. Section 2.6. in [Val03]). Now we let GLtn(Z) (cf. 4.1.2) act on the set of T -secondary cones by U.∆ := UT∆U. We call two T -secondary cones ∆,∆ 1 (T , t)-equivalent if there is U P GLtn(Z) such that UT∆U = ∆ 1 and UTTU = T . If Λ is a lattice we simply speak of T -equivalence, if T = Sn we simply speak of t-equivalence or GLtn(Z)-equivalence. 76 6.1. VORONOI’S SECOND REDUCTION THEORY A special class of subspaces T of Sn are those that are of the form T = TG, where G is a finite subgroup of GLtn(Z) and TG := tQ P Sn | UTQU = Q @U P Gu. In this case we will use the notation of (G, t)- equivalence synonymous for (TG, t)-equivalence andG-secondary cone synonymous for TG-secondary cone. Two Delone subdivisions D,D 1 are referred to as bistellar neighbors if their corresponding (T , t)- secondary cones are contiguous, that is if F := ∆T (D)X∆T (D 1) is facet. The transition from D to D 1 is then called a T -flip. 6.1.2 A polyhedral description of secondary cones The classical case of Voronoi’s second reduction theory for lattices was generalized to consider linear subspaces T Ă Sn (cf. [DSSV08]) and to cover periodic sets (cf. [Sch09], this source also discusses the result on linear subspaces). We recall the polyhedral description of T -secondary cones as given in [Sch09, Chapter 4]. This combines Theorems 4.3, 4.5 and 4.7 of the aforementioned work. To an element w P Rn and an affine basis V Ă Rn of Rn we associate the quadratic form NV ,w := ww T ´ ÿ vPV αvvv T , with αv being the affine coordinates ofwwith respect to the basis V . If V = tv1, . . . , vn+1u is the vertex set of a Delone simplex D, and w is such that D 1 = convtv2, . . . , vn+1,wu is another Delone simplex adjacent (facet-sharing) to D, we write ND,D1 instead of NV ,w. Let D be a polyhedral subdivision of Rn, such that Λ := vert(D) is a standard periodic vertex set (e.g. a Delone subdivision of any standard periodic vertex set). Let E denote the system of linear equations of the form xNV ,w,Qy = 0, where for each n-polytope D P D we choose the vertex set V Ă vert(D) of an arbitrary n-simplex S in D and for all w P vert(D) obtain an equation of the above form. Note that the above equations are trivially satisfied for all Q P rSně0 whenever D P D is itself a simplex (or more generally if w P V), and are therefore redundant in this case. Let U denote the null space of the system E on rSně0. Then the last comment in particular implies that if D is a triangulation, U = rSně0. Since NV+v,w+v = NV ,w for all v P Rn, and by periodicity of Λ, there are only finitely many such equalities. Furthermore let I denote the system of linear inequalities of the form xNVYw,w1 ,Qy ą 0, where for every facet F of the complex D and its adjacent n-polytopes D,D 1 P D with F = DXD 1 we choose the vertex set V Ă vert(F) of an arbitrary n´ 1-simplex S in F and for arbitrary w P vert(DzF), w 1 P vert(D 1zF) we obtain one inequality as above. The definition of U implies that the orthogonal projection piU(NVYw,w1) does not depend on the choice of V ,w,w 1 satisfying the above restrictions. Again we see that there are only finitely many distinct inequalities. The main theorem of Voronoi’s second reduction theory can than be formulated as follows. 77 6. DV-CELLS AND NORMALISED SECOND MOMENTS THEOREM 6.1.1. Let D be a polyhedral subdivision of Rn, such that Λ := vert(D) is a standard periodic vertex set and let T be a subspace of Sn. i. The closure ∆T (D) is a polyhedral cone in rSně0 X T and ∆T (D) = tQ P UX T | Q satisfies the system Iu. The system I and the number of defining equalities for U are finite. ii. The map D ÞÑ ∆T (D) gives an isomorphism between the poset of Delone subdivisions of Λ ordered by coarsening and the poset of closures of T -secondary cones ordered by inclusion. The closures of all T - secondary cones if Delone subdivisions form a polyhedral subdivision of rSně0 X T . PROOF. This is an amalgamation of Theorems 4.5, 4.7, 4.13 in [Sch09]. REMARK 6.1.2. The classical case of lattices goes back to Voronoi (cf. [Vor08]), the generalization to periodic sets is due to Schürmann (cf. [Sch09, Chapter 4]). We have used the language of the latter to formulate these results.  There are several finiteness results, that ensure the existence of a fundamental domain for the action of GL(Λ) = GLtn(Z), that is a finite union of secondary cones, if Λ is a periodic vertex set. We will briefly collect these here. THEOREM 6.1.3 (VORONOI’S SECOND REDUCTION THEORY). LetΛ = Ť tPt t+Zn be a standard periodic vertex set and G be a finite subgroup of GL(Λ). i. The topological closures of (G, t) secondary cones give a polyhedral subdivision of rSně0 X TG. The closures of two secondary cones have a common facet if and only if they are bistellar neighbors. ii. If Λ is rational, there are only finitely many (G, t)-inequivalent G-secondary cones. PROOF. This is an amalgamation of Theorems 4.7, 4.13, 4.19 in [Sch09]. REMARK 6.1.4. This includes the classical case with t = (0) and G being trivial. This can be made into an algorithm as described in [Sch09, Chapter 4], cf. [SVG] for an implementation of the lattice case.  6.1.3 Representatives of low-dimensional Delone triangulations of Zn There has been a great deal of work done on the classification of inequivalent Delone triangulations and subdivisions. This includes Voronoi’s classification of all inequivalent Delone triangulations up to dimension 4 (cf. [Vor08]). The classification of the 222 inequivalent Delone triangulations in dimension 5 was then settled by Engel and Grishukhin in 2002 (cf. [EG02]), building up on work of Ryshkov and Baranovskii (cf. [BR73], [RB78]) who already found 221 inequivalent Delone triangulations and work of Engel (cf. [Eng98]) who found the missing triangulation. 78 6.2. ON THE DV-CELL OF A MONOHEDRAL PERIODIC SET 6.2 ON THE DV-CELL OF A MONOHEDRAL PERIODIC SET Within this section (Λ,Q) will be a monohedral periodic set on (V ,Q), and without loss of generality we assume that 0 P Λ. It then suffices to study one particular Dirichlet-Voronoi cell for geometrical insight. We will work with DV(Λ,Q) := DVΛ,Q(0). To compute the explicit quantizer constant of a given lattice, we need knowledge about the geometry of its DV-cell. There are some rather classical results on the computation of Dirichlet-Voronoi cells and the quantizer constant (cf. [CS82], or Chapter 21 in [CS98]), and there is a more recent algorithm for these problems, which can handle higher dimensional lattices, provided a large enough amount of symmetry (cf. [DSSV09]). We will work towards a description the Dirichlet-Voronoi cell of a symmetric periodic set, including the case of lattices, which is suited quite well to find at least locally an explicit expression for the quantizer constant. The main obstacle to overcome is to handle the geometry of classes of periodic sets at once, we will now show how Voronoi’s second reduction theory aids to this cause. 6.2.1 The vertices of the DV-cell of a monohedral periodic set We argued earlier that we can replace an arbitrary periodic set by a coordinate representation through periodic forms (cf. Propositions 4.1.1 and 4.1.2). Of course this reduces to the case of quadratic forms in the case of lattices. So fix a standard periodic point set Λt Ă Rn. Let Q P Sną0 and denote D = DelΛt(Q). Then the DV polytope of t0u with respect to Λt and Q is determined through star(0,D) by means of its vertices as we noted in Lemma 4.3.17. Therefore knowledge about the induced Delone subdivision of a periodic form yields a possibility to explicitly compute its DV-cell: recall that for an n-simplex S Ă Rn with vert(D) = ts0, . . . , snu its centroid with respect toQ is the unique solution to the system of linear equalities given by |x´ si|Q = |x ´ s0|Q, where i P t1, . . . ,nu. If M(S) denotes the matrix whose i-th row is given by (si ´ s0)T , we obtain: (x´ si)TQ(x´ si) = (x´ s0)TQ(x´ s0) ô(si ´ s0)TQx = 12 (|si| 2 Q ´ |s0|2Q). Let a be the vector whose i-th entry is 12 |si´s0|2Q, that is, the vector of halves of squared norms of rows inM(S) and let b be the vector whose i-th entry is given by 12 (|si| 2 Q ´ |s0|2Q). With this notation c = s0 +Q ´1M(S)´1a = Q´1M(S)´1b (6.1) is the solution of the above system and thus the centroid of S. Now if D P star(0,D) is not a simplex, we can choose any n + 1 affinely independent vertices of D and compute the centroid of the simplex obtained in this way. Since D is a Delone polytope this actually is the centroid of D. 6.2.2 On a triangulation for Dirichlet-Voronoi polytopes We are interested in triangulations of Dirichlet-Voronoi cells of monohedral periodic sets, that do only depend on the associated secondary cone and are pyramidal with apex 0. Furthermore we would like 79 6. DV-CELLS AND NORMALISED SECOND MOMENTS a triangulation as such to add no new vertices, like for example the barycentric triangulation would do. This latter condition is posed to reduce the complexity of computations, by reducing the number of vertices involved. This of course comes with a trade-off: the barycentric triangulation might allow to exploit symmetries of the Dirichlet-Voronoi cell for later applications in computing, for example, the quantizer constant. For single lattices, opposed to secondary cones, exploiting symmetry by using a barycentric triangulation, lead to certain precise estimations of the quantizer constant of well-known lattices, that were inaccessible before (cf. [DSSV09]). The isomorphy given by Lemma 4.3.14 implies that it suffices to find a triangulation as such for the Dirichlet-Voronoi polytope corresponding to the Delone polytope t0u. We use the duality between Dirichlet-Voronoi and Delone subdivisions to define a triangulation, for which the vertices of the simplices involved can be labeled by a fixed enumeration of the elements of the star of t0u. Then a triangulation through pulling (cf. Lemma 4.2.1) does the trick. Let P be a Delone set with corresponding secondary cone ∆ and with Delone subdivision D. Fix an ordering of the full dimensional Delone polytopes in star(p,D), say (D1, . . . ,Dk). Let p P P. We define a lattice (order-theoretic) in 2t0,1,...,ku as follows: i. star(p,D) gives rise to a poset dual to the face lattice FQ for DVP,Q(p) for arbitrary Q P ∆o: we identify an element of star(p,D) by the n-dimensional Delone polytopes it is a common face of. Since the n-dimensional Delone polytopes are in bijection with their labels, we can therefore express any face by a subset of t1, . . . ,ku. Inclusion of faces in star(p,D) then induces an ordering relation on the so obtained subset of 2t1,...,ku, which is ordinary inclusion. By Proposition 4.3.15 we immediately see that this is in fact a lattice and that it is dual to the face lattice of DVP(Q, tpu) and thus to DVP,Q(p), since they are affinely isomorphic (by Lemma 4.3.14). ii. Since FQ = FQ1 for all Q,Q 1 P ∆o, we simply write F, it is the face lattice of a realizable convex polytope. iii. F and the ordering (D1, . . . ,Dk) induce the pulling triangulation T = T(D, (D1, . . . ,Dk),p) := p´k . . .p ´ 1 p ´ 0 (BF), cf. Lemma 4.2.2. PROPOSITION 6.2.1. Let P be a Delone set with corresponding secondary cone ∆ and with Delone subdivision D. Fix an ordering of the full dimensional Delone polytopes in star(p,D), say (D1, . . . ,Dk). Let p P P, and let T = T(D, (D1, . . . ,Dk),p) be as above. Define a map σQ by t ÞÑ # conv(tcentroidQ(Di) | i P tu) 0 R t conv(tpu Y tcentroidQ(Di) | 0 ‰ i P tu) 0 P t. Then i. σQ(T) is a pyramidal triangulation with apex p of DVP(p). ii. |t| ´ 1 is equal to the rank of σQ(t) for all t P T(D, (D1, . . . ,Dk),p) if and only if Q P ∆o. PROOF. Ad (i): We have seen in (iii) of Lemma 4.2.2, that for every polytope that realizes F, we obtain a triangulation, pyramidal with apex p, as desired. This proofs the claim for all Q P ∆o. If Q P B∆ 80 6.2. ON THE DV-CELL OF A MONOHEDRAL PERIODIC SET nothing to bad happens: σQ(T) still is a triangulation of DVP,Q(p), we only lose injectivity of the map σQ, non-maximal simplices in σQ(T) might have distinct preimages in T. Ad (ii): LetQ P ∆o and let t be an element of T(D, (D1, . . . ,Dk),p). By construction σQ(t) is a simplex. Assume that 0 R t, thus p R σQ(t). Let there be any labels i ‰ j, for which vi, vj P σQ(t). By definition vi, vj are centroids of Delone polytopes Di ‰ Dj in D thus implying vi ‰ vj. Therefore σQ(t) is a simplex with |t| different vertices, therefore its rank is |t| ´ 1. If 0 P t, the same is true by switching to tzt0u, since for all Di, the centroid vi is distinct from p. If on the other hand Q P B∆ we have that DelP(Q) is a coarsening of D and we therefore find a set of labels i1, . . . , ir such that eachDij no longer is a Delone polytope, but their join is. Any t containing at least two of the above labels therefore maps to the convex hull of at most |t| ´ 1 different vertices of DVP,Q(0), therefore the rank of σQ(t) is at most |t| ´ 2. In the implementation used for purposes of this thesis we chose to compute a pulling triangulation of C(BP) first and then build the pyramids over 0, this was done to decrease the dimension of polytopes on which the triangulation algorithm has to work on. ALGORITHM 6.2.2 (LOCAL VORONOI TRIANGULATION). i. Initialize D0 and fix an ordering (D1, . . . ,Dk) of its elements. ii. Compute the incidence poset of star(0,D) under the chosen labeling as in the proof of Proposition 6.2.1 and dualize it. Denote the dualized poset by F. iii. For every facet F P F obtain the face lattice F|F of F by restriction of F and compute the pulling triangulation p´k . . .p ´ 1 (F|F). iv. For each of the obtained triangulations build the pyramids over 0. v. The set of all of the above obtained pyramids gives an abstract triangulation T(D, (D1, . . . ,Dk)) as was proposed by Proposition 6.2.1. Another thing we will need to deal with is the orientation of the simplices in a triangulation of DVP,Q(0) induced by some T as in Proposition 6.2.1. The good news is: given Q,Q 1 P ∆o corre- sponding simplices σQ(t),σQ1(t) for t Ă T are equally oriented. LEMMA 6.2.3. Let P be a Delone set, p P P and T = T(D, (D1, . . . ,Dk),p) be as in Proposition 6.2.1. Let  P t˘1u, Q P ∆o and let σQ(t) with t P T(n) be -oriented. Then σQ1(t) is -oriented for all Q 1 P ∆o and for those Q 1 P B∆ where it is full dimensional. PROOF. We show this for the case p = 0, the general case follows by application of a suitable isometry. Write SQ = σQ(t) and SQ1 = σQ1(t). Consider the map Σ : ∆ Ñ Rnˆn given by Q ÞÑ M(SQ), where M(SQ) is the n by n matrix whose j-th column is given by the j-th vertex of SQ (excluding the zeroth vertex 0). This map is entrywise polynomial in the entries of Q and thus continuous, as is easily seen by invoking the explicit term (6.1) for these vertices, which by construction are centroids of a Delone simplex. Each of these centroids depends continuously on a change in Q, as Q´1 and the norms } ¨ }Q do. Therefore the map det ˝Σ : ∆Ñ R is continuous. 81 6. DV-CELLS AND NORMALISED SECOND MOMENTS Now SQ is -oriented if and only if sign(det(M(SQ))) = , that is, by definition, if sign(det ˝Σ(Q)) = . If Q 1 P ∆, we can move from Q to Q 1 along the line segment QQ 1 = tλQ + (1 ´ λ)Q 1 | λ P [0, 1]u as ∆ is convex. Now we claim that for arbitrary rQ P QQ 1zQ 1 we have that S rQ is -oriented. In any case, QQ 1zQ 1 Ă ∆o. Therefore, for any rQ P QQ 1zQ 1 we have that S rQ is full dimensional (cf. Proposition 6.2.1, (ii)) and det ˝Σ(rQ) ‰ 0. By continuity of det ˝Σ it follows that sign(det ˝Σ(rQ)) = . Since Q 1 is the limit of a sequence in QQ 1 we get sign(det ˝Σ(rQ)) P t0, u. If Q 1 P ∆o and therefore SQ1 is full dimensional we obtain that Q 1 is -oriented. If Q 1 P B∆ and SQ1 is full dimensional the above argument remains valid. 6.2.3 Classical results on DV-cells of lattices For sake of completeness we state some of the above mentioned classical results. DEFINITION 6.2.4 (RELEVANT POINT). A point q P P is called Dirichlet-Voronoi-relevant (DV- relevant or just relevant) for p P P (or for DV(p)), if the hyperplane between p and q contains a facet of DV(p). ˛ In the case of a lattice Lwe will call a lattice point relevant if it is a relevant point of DV(L). The following Theorem is due to Voronoi, and characterizes the relevant points of any lattice L. THEOREM 6.2.5 ( [CS98], CH. 21, THEOREM 10). A point 0 ‰ v P L is relevant if and only if ˘v are the only shortest vectors in the coset v+ 2L. It is not surprising that the following lemma holds: LEMMA 6.2.6 (MINIMAL VECTORS ARE RELEVANT). If v P Min(L), then v is relevant. PROOF. We have to assure that ˘v are the only vectors of length ‖v‖2 in v + 2L. Therefore let v ‰ u P v+ 2L: u = v+ 2w, w P L. Then ‖u‖2 = ‖v‖2 + 4‖w‖2 + 4b(v,w) If ‖u‖2 = ‖v‖2, this leads to ´b(u,w) = ‖w‖2. By the Cauchy-Schwarz inequality we have b(v,w)2 ď ‖v‖2 ¨ ‖w‖2 with equality only if v,w are linearly dependent. In that case we obtain ‖v‖2 = ‖w‖2 which only is possible for w = ´v, therefore u = ´v. If v,w are linearly independent we have ‖w‖2 ă ‖v‖2, which would contradict v P Min(L). 6.3 A LOCALLY EXACT FORMULA AND A CONSTRAINED POLYNOMIAL OPTIMIZATION PROGRAM FOR THE LATTICE VECTOR QUANTIZER PROBLEM The quantization problem is a minimization problem, where the objective function does not come in a handy way to employ standard optimization theory, such as linear or semi-definite programming. We will turn the problem of finding the optimal lattice quantizer of a given dimension n into a collection of constrained minimization problems, for each of which the objective function, as well as the constraints are polynomial functions. This is done with the help of Voronoi’s second reduction theory, the same approach was used in [SV06] for the lattice covering and lattice covering packing problem. 82 6.3. A LOCAL FORMULA AND POLYNOMIAL PROGRAM FOR THE QUANTIZER PROBLEM We now assume to be in a more restricted version of the setup of Theorem 6.1.3. This will be the general assumption on periodic sets for the application of Voronoi’s second reduction theory to the periodic quantizer problem. ASSUMPTION 6.3.1. We assume that Λt is a rational standard periodic vertex set such that there exists a finite G ă GL(Λ) = GLtn(Z) for which (Λt,Q) is monohedral for all Q P Q P Sną0 X TG.  An example for non-lattices that satisfy Assumption 6.3.1 are periodic sets that are the join of 2 trans- lates of a lattice (cf. Lemma 4.3.6). The general periodic quantizer problem is a problem of periodic sets in Euclidean standard space, but every such set is isometric to a coordinate representation, as we noted in Proposition and Definition 4.1.2. In particular, we identify lattices with positive definite quadratic forms by fixing the set of lattice points to be the standard vertex set Zn (Proposition and Definition 4.1.1). For lattices the group G above can be chosen to be trivial, lattices are symmetric by virtue of the action of translations and thus regardless of the involved quadratic form. Let L Ă Rn be a lattice with respect to the standard inner product and associated coordinate lattice (Zn,Q). We are interested in minimizing G(L) = det(L)´1/2´1/n ş DV(L) }x}2dx. Since we prefer to work with the coordinate representation of L by (Zn,Q), we reformulate this expression in terms ofQ. We, more generally, derive this for periodic sets. LEMMA 6.3.2. Let Λ = AΛt Ă Rn be a symmetric m-periodic set with respect to the standard inner product and associated coordinate representation (Λt,Q), that is Q = ATA. Then G(Λ) = G(Q, t), where G(Q, t) := m 1+2/n n det(Q)´1/n ş DV(Q,t) }x}2Qdx. PROOF. This becomes evident by application of the formula of change of variables for integralsż DV(Λt,Q) }x}2Qdx = ż DV(Λt,Q) }Ax}2dx = det(A)´1 ż ADV(Λt,Q) }x}2dx = det(A)´1 ż DV(Λ) }x}2dx, as DV(Λ) = ADVΛt(Q) (recall that DV(Q, t) = DV(Λt,Q)). A standard technique to compute ş DV(Λ) }x}2dx is to find a triangulation T of DV(Λ) and computeş DV(Λ) }x}2dx = ř SĂT dim(S)=n ş S }x}2dx via the following formula. THEOREM 6.3.3 ( [CS98], CH. 21, THEOREM 2.). Let V be the Euclidean standard space of dimension n, and let S Ă V be a simplex with vertices s0, . . . , sn and barycenter sˆ = 1n+1 řn i=0 si. Thenż S }x}2dx = vol(S) (n+ 1)(n+ 2) ¨ ( }(n+ 1)sˆ}2 + nÿ i=0 }si}2 ) . This then applies to ş DVZn(Q) }x}2Qdx as follows. 83 6. DV-CELLS AND NORMALISED SECOND MOMENTS COROLLARY 6.3.4. Let (V ,Q) be a quadratic space of dimension n with norm } ¨ }Q, and let S Ă V be a simplex with vertices s0, . . . , sn and barycenter sˆ = 1n+1 řn i=0 si. Thenż S }x}2Qdx = volQ(S)adet(Q) 1(n+ 1)(n+ 2) ¨ ( }(n+ 1)sˆ}2Q + nÿ i=0 }si}2Q ) = vol(S) (n+ 1)(n+ 2) ¨ ( }(n+ 1)sˆ}2Q + nÿ i=0 }si}2Q ) . PROOF. Write Q = ATA (Cholesky decomposition). Now by the formula for the change of variables for integrals we have:ż S }x}2Qdx = ż S }Ax}2dx = 1 det(A) ż AS }x}2dx = 1 det(A) ¨ vol(AS) (n+ 1)(n+ 2) ¨ ( }(n+ 1)Asˆ}2 + nÿ i=0 }Asi}2 ) , where vol(AS) = det(A) ¨ vol(S) and volQ(S) = a det(Q) ¨ vol(S), so that the claim is proven. Employing Voronoi’s second reduction theory we obtain the knowledge needed about DVΛt(Q): with respect to the rational standard periodic vertex set Λt we choose a representative set of secondary cones, which is finite by Theorem 6.1.3. For each of these secondary cones we will formulate a local quantizer problem, since we can simultaneously triangulate the Dirichlet-Voronoi cells associated to quadratic forms in a fixed secondary cone: Under the Assumption 6.3.1, the ∆-local quantizer problem is given by min G(Q, t) s. t. Q P ∆. From now on we fix a secondary cone ∆. Employing Lemma 4.3.17 we can compute the DV-cell of any monohedral periodic set (Λt,Q) for Q P ∆ in a general way. Using Proposition 6.2.1 we can find a triangulation for each of these lattices that can be described entirely by the means of ∆ itself. LEMMA 6.3.5. Let S1, . . . ,Sn Ă Rn be simplices and denote the centroid of Si by ci. Suppose the simplex S := convt0, c1, . . . , cnu is full dimensional. Let Q P Sną0 and let rQ be the adjoint of Q, that is, QrQ = det(Q)In. i. The quantity det(Q)}ci}2Q = }Qci}2rQ is polynomial in the entries of Q. ii. For λ1, . . . , λn P R the quantity det(Q)}řni=1 λici}2Q = }řni=1 λiQci}2rQ is polynomial in the entries of Q. For the polynomial expressions above, the coefficients are determined solely by S1, . . . ,Sn. PROOF. We have already seen that for Q P Sną0 and S Ă Rn a simplex, its centroid is given by c = Q´1M(S)´1b. Thus }c}2Q = (Q´1M(S)´1b)TQ(Q´1M(S)´1b) = bT (A´1)TQ´1M(S)´1b = 1 det(Q) bT (M(S)´1)T rQM(S)´1b = 1 det(Q) }Qc}2rQ, 84 6.3. A LOCAL FORMULA AND POLYNOMIAL PROGRAM FOR THE QUANTIZER PROBLEM where the entries of b (and therefore Qc =M(S)´1b) are (linear) polynomials in the entries of Q, and the entries of rQ are polynomial of degree n´1 in the entries ofQ since each entry is a first minor ofQ. Thus }c}2Q is rational in the entries of Q and det(Q)}c}2Q is polynomial in the entries of Q, both are of degree n+ 1. This proves the first assumption. For λ1, . . . , λn P R and řni=1 λici = Q´1 řni=1 λiA´1i bi analogously det(Q)}řni=1 λici}2Q is polynomial of degree n+ 1. This shows that if S Ă T is a simplex in a triangulation of DVΛt(Q), the norms of its vertices si are given by rational expressions in the entries of Q, as is the norm of their barycenter ps. To be precise we have: }si}2Q = 1det(Q)}Qsi}2rQ and }ps}2Q = 1det(Q)}Qps}2rQ, where }Qsi}2rQ and }Qps}2rQ are polynomial in the entries of Q. The volume of a n-simplex S Ă Rn can be calculated in different ways, rather then employing Cayley- Menger determinants, we use the volume of a related parallelepiped. LEMMA 6.3.6. Let ∆ = ∆(D) be a fixed secondary cone and fix an ordering (D1, . . . ,Dk) for the elements of star(0,D). Let T = T(D, (D1, . . . ,Dk), 0) be a triangulation as obtained by Proposition 6.2.1. For an n-simplex S Ă Rn with vert(S) = ts0, s1 . . . , snu letM(S) be the matrix whose j-th column is sj ´ s0. Let t P T be maximal element. Then there exists  = (t) P t˘1u such that for arbitrary Q P ∆ volQ(σQ(t)) =  1 n! det(Q)1/2 det(M(σQ(t))), and vol(σQ(t)) =  1 n! det(M(σQ(t))), where det(Q)1/2 ¨ volQ(σQ(t)) and det(Q) ¨ vol(σQ(t)) are polynomial in the entries of Q. PROOF. Note that from the construction of the triangulation, M(σQ) composes of the nontrivial ver- tices of σQ, since the zeroth vertex always is 0. Then vol(σQ(t)) = ˇˇˇˇ 1 n det(M(σQ(t))) ˇˇˇˇ =  1 n det(M(σQ(t))) =  1 n det(Q)´1 det(Q ¨M(σQ(t))), where  is the orientation of σQ(t), but this only depends on t and ∆ and not on the choice of Q in ∆, as we proved in Lemma 6.2.3. The vertices si of σQ(t) are of the form si = Q´1A´1i bi, where si is the centroid of a Delone simplex Di P star(0,D). Now det(Q ¨M(σQ(t))) is a polynomial in the entries ofQ, as we already noted in the proof of Lemma 6.2.3. Thus the volume of σQ(t) is rational in the entries of Qwhere the denominator det(Q) comes from the factor det(Q)´1 in vol(σQ(t)) =  1n det(Q) ´1 det(Q ¨M(σQ(t))). Now we are prepared to find an objective function that is polynomial in the entries of Q and can be used to solve the quantizer problem at least ∆-locally for each secondary cone ∆. The constraints we need areQ P ∆which can be expressed by linear (and therefore polynomial) inequalities (cf. Theorem 6.1.1) and det(Q) ě 1. The latter one is of course polynomial in the entries of Q and is needed to avoid the remainder det(Q)e that comes from the norms of the vertices, as well as the volume of the simplices in D0. 85 6. DV-CELLS AND NORMALISED SECOND MOMENTS THEOREM 6.3.7. Let Λt satisfy Assumption 6.3.1. Let ∆ be a secondary cone for Λt. The ∆-local quantizer problem min G(Q, t) s. t. Q P ∆ is equivalent to min G∆(Q) s. t. Q P ∆ s. t. det(Q) ě 1 where G∆(Q) is a polynomial in the entries of Q. That is, the ∆-local quantizer problem is equivalent to a polynomial optimization problem with convex constraints; the constraint Q P ∆ can be replaced by a finite system of linear inequalities. PROOF. We have ż DVZn(Q) }x}2Qdx = ÿ SĂT(Q) dim(S)=n ż S }x}2Qdx. Using Corollary 6.3.4 the previous Lemmata 6.3.5 and 6.3.6 show that each of the righthand side sum- mands is the quotient of a polynomial in Q by det(Q)2: det(Q)2 ¨ ż S }x}2Qdx = det(Q)2 ¨ vol(S)(n+ 1)(n+ 2) ¨ ( }(n+ 1)sˆ}2Q + ÿ sPvertS }s}2Q ) = (S) ¨ det(Q) ¨ det(M(S)) (n+ 2)! ¨ ( }(n+ 1)Qsˆ}2rQ + ÿ sPvertS }Qs}2rQ ) . Substituting this we obtain G(Q) = 1 n det(Q)´1/n ż DVZn(Q) }x}2Qdx = det(Q)´1/n´2 ¨G∆(Q), where G∆(Q) := 1 n(n+ 2)! ÿ SĂT(Q) dim(S)=n (S) ¨ det(Q) ¨ det(M(S)) ¨ ( }(n+ 1)Qsˆ}2rQ + ÿ sPvertS }Qs}2rQ ) is polynomial in the entries of Q. Since the quantizer constant is invariant to scaling we can assume that det(Q) = 1 is fixed without missing an optimal solution, thus we get rid of the denominator. The relaxation det(Q) ě 1 poses no threat to finding optimal solutions, since G∆(λQ) = λ(1+2n)G∆(Q) and therefore replacing a Q with det(Q) ą 1 by λQ such that det(λQ) = 1 results in G∆(λQ) ă G∆(Q). Finiteness of the involved system of linear inequalities was observed in Theorem 6.1.1. 86 6.4. ON THE LATTICE QUANTIZER PROBLEM IN LOW DIMENSIONS REMARK 6.3.8. We chose to drop t from the notation for G∆, it is implicitly involved since ∆ is, by assumption, a secondary cone for Λt.  We state the following as a Corollary to have an explicit formula for the quantizer constant of a lattice at hand whenever needed. COROLLARY 6.3.9. Let Q P Sną0, and let ∆ be the secondary cone of DelZn(Q). Then G(Q) = G∆(Q) det(Q)1/n+2 , where G∆(Q) is a polynomial in the coefficients of Q. The coefficients of G∆(Q) depend only on ∆. REMARK 6.3.10. In the subsequent we will provide the quantizer polynomials for lattices up to dimension 4. In all of these cases it turns out that G∆(Q) is a multiple of det(Q) (the determinant of an abstract matrix Q P ∆ as a polynomial in its entries). This is an effect that occurs only for the full DV-cell, for an simplex the corresponding summand will in general not be divisible by det(Q), as can be readily checked using the MAPLE-code described in Appendix B.  6.4 ON THE LATTICE QUANTIZER PROBLEM IN LOW DIMENSIONS 6.4.1 Introductory remarks Conic parameters for quadratic forms If we fix a secondary cone ∆ we know that each Q P ∆ can be written as a conic combination Q =ř k λk(Q)R (k) where the R(k) are the extreme rays of∆. for a givenQwe set c(Q) := (λ1(Q), . . . , λm(Q)) to be the vector of conic parameters. Now evidently we can transform any quantity F, which is given explicitly in the entries of Q, into a quantity which is given in the conic parameters of Q, simply by substituting qij = ř k λk(Q)r (k) ij . We denote such a transformation by F(c). There is another similar kind of parameters for quadratic forms, the Selling parameters: Let Q = (qij)ni,j=1. We write qij := e T iQej for i, j P t1, . . . ,n + 1u, where en+1 = ´ nř i=1 ei. Thenřn+1 j=1 qij = 0, and therefore qii = ´ řn+1 i‰j=1 qij. Then we call the parameters qij, for i, j P t1, . . . ,n + 1u, i ‰ j, the Selling parameters of Q. Below we will discuss their relation to the conical parameters in a special case. Voronoi’s principal domain of the first type A common example of a Delone triangulation in dimension n is given by Voronoi’s principal domain of the first type, given by ∆(Vn): Let Q P Sną0 be given by qii = n, qij = ´1 for all i, j P t1, . . . ,nu, i ‰ j. The lattice (Zn,Q) is then isometric to A#n. We find Vn := DelZn(Q) = tDσ + v | σ P Sn+1, v P Znu, 87 6. DV-CELLS AND NORMALISED SECOND MOMENTS where Dσ = convteσ(1), eσ(1) + eσ(2), eσ(1) + . . . + eσ(n+1)u, with the convention en+1 = ´řni=1 ei. The automorphism group of Vn is isomorphic to Sn+1. We will use the snake notation of Ryshkov, to write simplices as Dσ above in a more condensed way: xv1, . . . , vny = convtv1, v1 + v2, . . . , v1 + . . . + vnu. (6.2) Thus Dσ = xeσ(1), . . . , eσ(n+1)y. One finds (cf. §102´ 104 in [Vor08]) ∆(Vn) = tQ P Sn | qij ă 0 for i ‰ j and nÿ i=1 qij ą 0 for i P t1, . . . ,nuu. The extreme rays of this cone are the matrices E0i for 1 ď i ď n and Eij for 1 ď i ă j ď n. Here E0i is such that the entry at position (i, i) is equal to 1, while all other are equal to 0 and Eij is such that the entries at positions (i, i), (j, j) are equal to 1, the entries at positions (i, j), (j, i) are equal to ´1, and all other entries are equal to 0. IfQ P ∆, we write λ0i for the coefficient of E0i and λij for the coefficient of Eij. The Selling parameters and conical parameters of Q P V2 satisfy the following relation: qij = # ´λij for i, j ď n, ´λ0i for i ď n, j = n+ 1. 6.4.2 Lattice quantizer polynomials and globally optimal solutions in dimensions 2 and 3 Dimension 2 We use this case to give a more detailed example of how to compute the quantizer polynomial, every- thing of importance can be seen here already, while the presentation of the discussion is quite simple. There is only one equivalence class of Delone triangulations in dimension 2, this was already settled by Voronoi (cf. [Vor08]), it can be checked by computer using the algorithm sketched in the book [Sch09], cf. [SVG]. A representative is given by Voronoi’s principal domain of the first type (cf. 6.4.1). The secondary cone ∆ = ∆(V2) can then be described as follows: ∆ := ∆(V2) = ! Q P S2 qij ă 0 for i ‰ j and ř2i=1 qij ą 0 for j = 1, 2 ) = Q P S2 q12 ă 0 and q11 + q21 ą 0 and q21 + q22 ą 0 ( . Those maximal cells of the triangulation V2, that contain the origin 0, are given by star(V2, 0). In hindsight of our wish to fix a lattice in 2t0,1,...,6u which produces a simultaneous triangulation of any corresponding Dirichlet-Voronoi cell, we fix an order on the elements of V20. Since we can index these 88 6.4. ON THE LATTICE QUANTIZER PROBLEM IN LOW DIMENSIONS elements by elements of the symmetric group S3 it seems rather natural to start from there. A seem- ingly natural way to numerate the elements of S3 is to index them by the order in which the Steinhaus- Johnson-Trotter algorithm produces them. For S3 this amounts to 1 fl id, 2 fl (23), 3 fl (132), 4 fl (13), 5 fl (123), 6 fl (12). Then star(V2, 0)(2) = tD1, . . . ,D6u, where D1 = Did = convt0, e1, e1 + e2u, D2 = D(23) = convt0, e1,´e2u, D3 = D(132) = convt0,´e2,´e1 ´ e2u, and D4 = ´D1,D5 = ´D2,D6 = ´D3. Now Lemma 4.3.17 shows that the vertex set of the Voronoi cell of an arbitrary Q P ∆ X S2ą0 can be read of from V20 by vi = centroidQ(Di). However, we should note that for Q on the boundary of ∆ certain of these points vi will coincide, this is a consequence of the fact that there are usually more inequivalent Delone subdivisions than triangulations, but those appear as the limiting cases of them. Here an example would be the standard form, where v1 = v2 and v4 = v5. In the case of a Q P ∆0 we have that DVZn(Q) is a permutahedron of order 3. The pulling triangulation that we described in Algorithm 6.2.2 coming from the labeling of elements of D0 as above leads to an polynomial expression in the entries of Q for the quantizer constant that computes to G(Q) = 1 24 ¨ q 2 11q22 + q11q 2 22 ´ 2q11q212 ´ 2q22q212 ´ 2q312 (q11q22 ´ q212)3/2 . (6.3) Since ∆ = cone "( 1 0 0 0 ) , ( 0 0 0 1 ) , ( 1 ´1 ´1 1 )* we can parametrize an arbitrary Q P ∆ by λ01, λ02, λ12 P Rě0 (cf. 6.4.1), the coefficients of Q as a conic combination of the extreme rays of ∆. This parametrization is closely related to the selling parameters of Q, they differ from one another by a change of sign, as noted above. Now as described in general above we can transform the quantity G(Q) to be an expression in c(Q) by substituting q11 = λ01 + λ12, q22 = λ02 + λ12 and q12 = ´λ12, this computes to G(c)(c(Q)) = 1 24 ¨ λ 2 01λ02 + λ 2 01λ12 + λ01λ 2 02 + λ01λ 2 12 + λ 2 02λ12 + λ02λ 2 12 + 4λ01λ02λ12 (λ01λ02 + λ01λ12 + λ02λ12)3/2 . (6.4) The related quantizer polynomial is given by G (c) ∆ (c(Q)) = λ01λ02 + λ01λ12 + λ02λ12 24 ¨ (λ201λ02 + λ201λ12 + λ01λ202 + λ01λ212 + λ202λ12 + λ02λ212 + 4λ01λ02λ12). THEOREM 6.4.1 (CF. P. 81 IN [FT53], P. 60 IN [CS98]). The optimal (lattice) quantizer in dimension 2 is given by any lattice arithmetically equivalent to the hexagonal lattice A2 – A#2, the optimal value is G(A2) = 5 108 ? 3. This is the only local minimum of the lattice quantizer problem in dimension 2. 89 6. DV-CELLS AND NORMALISED SECOND MOMENTS Dimension 3 As in dimension 2, there is only one equivalence class of Delone triangulations in dimension 3, a representative is given by Voronoi’s principal domain of the first type (cf. 6.4.1). The secondary cone ∆ can then be described as follows: ∆ := ∆(V3) = ! Q P S3 qij ă 0 for i ‰ j and ř3i=1 qij ą 0 for j = 1, 2, 3 ) that is the set of forms Q P S3 such that the coefficients satisfy q12 ă 0,q13 ă 0,q23 ă 0 q11 + q21 + q31 ą 0 q12 + q22 + q32 ą 0 q13 + q23 + q33 ą 0. Those maximal cells of the triangulation V3, that contain the origin 0, are given by star(V3, 0): star(V3, 0)(3) = tD1, . . . ,D24u, where again we numerate the elements of V30 by associating an permutation in S4 to the position it has in the list the Steinhaus-Johnson-Trotter algorithm produces. IfQ P ∆o we know that DVZ3(Q) is a permutahedron of order 4, i.e., a truncated octahedron. There are two combinatorially different types of facets appearing: 4-gons and 6-gons. A pulling triangulation as described in Algorithm 6.2.2 corresponding to the above ordering of vertices can be obtained by pulling each facet individually and building the pyramids with apex 0 above each simplex won by this. To be precise it suffices to execute simply one pulling refinement by the vertex with lowest index. If the facet is a 4-gon we obtain two simplices, if it is a 6-gon we obtain 4 simplices. It is possible to obtain the polynomial G(c)∆ (c(Q)) directly from this combinatorial description, cf. [BS83]. This leads to the a polynomial expression in the entries of Q for the quantizer constant that computes to G(Q) = G∆(Q) det(Q)7/3 (6.5) with G∆(Q) = det(Q) 36 ¨ (q211q22q33 ´ q211q223 ´ 2q11q212q33 + 4q11q12q13q23 ´2q11q213q22 + q11q222q33 ´ 2q11q22q223 + q11q22q233 ´2q11q323 ´ 2q11q223q33 ´ 2q312q33 + 2q212q213 + 6q212q13q23 ´2q212q22q33 + 2q212q223 ´ q212q233 + 6q12q213q23 +4q12q13q22q23 + 6q12q13q223 + 4q12q13q23q33 ´ 2q313q22 ´q213q222 ´ 2q213q22q33 + 2q213q223 ) 90 6.4. ON THE LATTICE QUANTIZER PROBLEM IN LOW DIMENSIONS Since ∆ = cone $&% 1 0 00 0 0 0 0 0  , 0 0 00 1 0 0 0 0  , 0 0 00 0 0 0 0 1  ,  1 ´1 0´1 1 0 0 0 0  ,  1 0 ´10 0 0 ´1 0 1  , 0 0 00 1 ´1 0 ´1 1 ,.- , we can parametrize an arbitraryQ P ∆ by λ01, λ02, λ03, λ12, λ13, λ23 P Rě0, the coefficients ofQ as a conic combination of the extreme rays of ∆. This parametrization is closely related to the selling parameters of Q, they differ from one another by a change of sign. Now as described in general above, and laid out for dimension 2 we can transform the quantityG∆(Q) to be an expression in c(Q), this computes to G (c) ∆ (c(Q)) = det(c) 36 ¨ ( det(c) ¨ Σ1 + 2 ¨ Σ2 + Σ3 ) , (6.6) where Σ1 = (6)ÿ λ01, Σ2 = (3)ÿ λ01λ02λ13λ23, and det(c) = (12)ÿ λ01λ02λ13 + (4)ÿ λ01λ02λ03. THEOREM 6.4.2 (THEOREM 1 IN [BS83]). The optimal lattice quantizer in dimension 3 is given by any lat- tice arithmetically equivalent to the lattice A#3 – D#3, the optimal value is G(A#3) = 19/(96 3 ? 16). This is the only local minimum of the lattice quantizer problem in dimension 3. 6.4.3 The lattice quantizer polynomials in dimension 4 There are 3 inequivalent secondary cones that represent the three inequivalent Delone triangulations in dimension 4. We use the description found in [Val03, 4.4.1]. From there we read off that we can describe those cones with the following positive semi-definite forms E01, . . . ,E04,E12,E13, . . . ,E34 together with Ea :=  2 1 ´1 ´1 2 ´1 ´1 2 0 2  ,Eb :=  1 1 ´1 ´1 1 ´1 ´1 1 1 1  where E0i shall be the all zero matrix except that the i-th diagonal entry is equal to 1 and Eij shall be the all zero matrix except that the i-th and j-th diagonal entries equal 1 and the entries at positions ij and ji equal ´1. Note that Ea represents the root lattice D4. We set ∆1 = ∆(Del(Q1)) = int (cone tE01, . . . ,E04,E12,E13, . . . ,E34u) (6.7) ∆2 = ∆(Del(Q2)) = int (cone tE01, . . . ,E04,E13, . . . ,E34,Eau) (6.8) ∆3 = ∆(Del(Q3)) = int (cone tE01, . . . ,E04,E13, . . . ,E24,Ea,Ebu) , (6.9) 91 6. DV-CELLS AND NORMALISED SECOND MOMENTS where Q1 =  4 ´1 ´1 ´1 4 ´1 ´1 4 ´1 4  ,Q2 =  5 1 ´2 ´2 5 ´2 ´2 6 ´1 6  ,Q3 =  6 2 ´3 ´3 6 ´3 ´3 6 1 6  . In particular ∆1 = ∆(V4). From this we can read of any data we need. The remaining 49 inequivalent Delone subdivisions (that are not triangulations) correspond to the faces of the closures of these cones, we therefore can compute their secondary cones from the representations in (6.7),(6.8),(6.9) above. From another look into [Val03, 4.4.1] we obtain that ∆1 shares each of its facets, and therefore each of its faces, with a GL4(Z) translate of ∆2 and ∆3 shares but one of its facets with a GL4(Z) translate of ∆2, where its remaining facet is shared with a GL4(Z) translate of itself. This comes from the fact that the corresponding Delone triangulations are bistellar neighbors of one another in exactly the same pattern. Therefore we can proceed by finding a GL4(Z) representative set of facets (and then on of faces of them) of ∆2 to cover the majority of cases. The remaining work then is to evaluate ∆1 itself, ∆3 itself and the remaining facet of ∆3 (and of course its faces). The 52 inequivalent secondary cones This classification has been achieved quite a few times, we refer to [Val03, 4.4.4 to 4.4.6] for an exposi- tory treatment. Though one can read off the 52 inequivalent secondary cones from a table in 4.4.6 of the above cited source, we go ahead and compute them ourselves by directly computing the rays of each of the repre- sentatives. We collect the data in a table; for more information we again refer to [Val03, 4.4.6]. Using this information it is possible to derive the remaining 49 quantizer polynomials from those given for the three inequivalent Delone triangulations in dimension 4, below. The local quantizer polynomial for ∆(V4): We start with the rational closure ∆1 of the secondary cone of the Delone triangulation V4 of Voronoi’s principal form of the first type. Those maximal cells of the triangulation V4, that contain the origin 0, are given by the star of V4 around 0. We set D11 :=ă e1, e2, e3, e4, e5 ą written in the snake notation of Ryshkov (cf. (6.2)), where e5 = ´e1 ´ e2 ´ e3 ´ e4. Then V40 is given by the S5 orbit of D11 where S5 is supposed to act on the subscripts of the vectors. This in particular implies that the automorphism group of V4 is isomorphic to S5. Thus star(V4, 0)(4) = tD11, . . . ,D1120u, where we numerate the elements of star(V4, 0)(4) by associating an permutation in S5 to the position it has in the list the Steinhaus-Johnson-Trotter algorithm produces. By the same reasoning as above we can read off the Dirichlet-Voronoi cell ofQ P S4ą0 from V40. IfQ P ∆o1 we know that DVZ4(Q) is a permutahedron of order 5. 92 6.4. ON THE LATTICE QUANTIZER PROBLEM IN LOW DIMENSIONS Table 6.1: Representative vectors of the 52 inequivalent secondary cones in dimension 4. dim Representative vector 1 (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) 2 (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) 3 (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0) 3 (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0) 4 (1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0) 4 (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0) 4 (1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0) 4 (1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0) 4 (0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0) 5 (1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0) 5 (0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0) 5 (0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0) 5 (1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0) 5 (1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0) 5 (1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0) 5 (0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0) 5 (1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0) 6 (1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0) 6 (0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0) 6 (1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0) 6 (1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0) 6 (1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0) 6 (1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0) 6 (0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0) 6 (1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0) 6 (1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0) 6 (0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0) dim Representative vector 7 (1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0) 7 (1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0) 7 (1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0) 7 (0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0) 7 (1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0) 7 (1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0) 7 (1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0) 7 (1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0) 7 (1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0) 7 (0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0) 7 (1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0) 8 (1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0) 8 (1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0) 8 (1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0) 8 (1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0) 8 (1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0) 8 (1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0) 8 (1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0) 9 (1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0) 9 (1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0) 9 (1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0) 9 (1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1) 10 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0) 10 (1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0) 10 (1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1) Recall that ∆1 = cone tE01, . . . ,E04,E12,E13, . . . ,E34u , so for i = 1, . . . , 4 we denote by λ0i = λ0i(Q) the coefficient of E0i in a conic representation of Q P ∆1, and for i = 1, . . . , 3, i ă j ď 4 with λij = λij(Q) the coefficient of Eij. S5 leaves G invariant by acting on the indices1 of the conical parameters, for if ρ P S5 we have that λab(ρ(Q)) = λρ(a)ρ(b)(Q). by choice of labeling. This explains our choice in naming the extreme rays and it allows to write up G (c) ∆1 in a condensed form, which we chose to resemble the one for the three dimensional case V3: G (c) ∆1 (c(Q)) = det(c) 48 ¨ ( det(c) ¨ Σ1 + Σ2 + 3 ¨ Σ3 + Σ4 ) , (6.10) 1We assume that the permutation acts on the unordered set of subscripts, these of course uniquely determine the parameter. 93 6. DV-CELLS AND NORMALISED SECOND MOMENTS where Σ1 = ÿ(10) λ01, Σ2 = ÿ(60) λ01λ02λ03λ12λ14 + ÿ(60) λ01λ02λ03λ12λ34 + 2 ¨ ÿ(60) λ01λ02λ03λ14λ24, Σ3 = ÿ(12) λ01λ02λ13λ24λ34, Σ4 = ÿ(30) λ01λ02λ03λ04λ12, and finally det(c) = ÿ(5) λ01λ02λ03λ04 + ÿ(60) λ01λ02λ03λ14 + ÿ(60) λ01λ02λ13λ24. The local quantizer polynomial for ∆2: We continue with the rational closure ∆2, let D2 denote its associated delone triangulation. Those maximal cells of the triangulation D2, that contain the origin 0, are given by the star of D2 around 0. Following [Val03, 8.4.2] we set D2I := ă e1, e2 ´ e1, e1 + e5, e4, e3 ą D2VIII := ă e1, e5, e2, e4, e3 ą D2II := ă e1, e2 ´ e1, e1 + e3, e4, e5 ą D2XI := ă e1, e3, e2, e5, e4 ą D2V := ă e1, e2 ´ e1, e1 + e5, e3, e4 ą D2IV := ă e1, e5, e2, e3, e4 ą D2VI := ă e1, e2 ´ e1, e1 + e4, e3, e5 ą D2XII := ă e1, e4, e2, e3, e5 ą D2IX := ă e1, e2 ´ e1, e1 + e4, e5, e3 ą D2VII := ă e1, e4, e2, e5, e3 ą D2X := ă e1, e2 ´ e1, e1 + e3, e5, e4 ą D2III := ă e1, e3, e2, e5, e4 ą which follows Voronois numeration (cf. p. 169 in [Vor08]). Again we use the snake notation of Ryshkov (cf. (6.2)), where e5 = ´e1 ´ e2 ´ e3 ´ e4. IfD is any of the above simplices we denote its image under ´ id by the roman numeral that corresponds to 12 + iwhere i is the value of the roman numeral of D. star(0,D2)(4) then consists of the translates of the above numbered 24 simplices by their vertices. This amounts to a total of 120 elements. Thus star(0,D2)(4) = tD21, . . . ,D2120u, where we numerate the elements of star(0,D3)(4) by starting with, say, D21 := D 2 I , followed by its translates D22, . . . ,D 2 5 and continuing with D 2 6 being D 2 II and so on. We compute Aut(∆2) – Di6 to be isomorphic to the dihedral group of order 12. In fact we compute the matrix group fixing Ea, which is isomorphic to Aut(D4) and afterwards the subgroup fixing ∆2. This matrix group is isomorphic to the direct product of the dihedral group of order 12 and a cyclic group of order 2 acting on the extreme rays with the cyclic factor being the kernel of the operation. Aut(∆2) acts on the conical parameters if we write it as subgroup of S5, to be explicit we use the subgroup generated by (03),(04), and (12). By abuse of language we will also refer to this group as Aut(∆2). Doing this we can derive a “compressed” expression for G(c)∆2 , however it is unfortunately still quite messy: G (c) ∆2 = det(c) 240 ¨ ( 5ÿ i=0 λia ¨ Σi ) , (6.11) 94 6.4. ON THE LATTICE QUANTIZER PROBLEM IN LOW DIMENSIONS where Σ5 = 104, Σ4 = 130 ¨ (ÿ(6) λ01 + ÿ(3) λ03 ) , Σ3 = 20 ¨ (ÿ(6) λ201 + ÿ(3) λ203 ) + 130 ¨ (ÿ(12) λ01λ03 + ÿ(6) λ01λ13 + ÿ(6) λ01λ23 + ÿ(3) λ03λ04 ) + 160 ¨ (ÿ(6) λ01λ34 + ÿ(3) λ01λ02 ) , Σ2 = 15 ¨ (ÿ(12) λ201λ03 + ÿ(12) λ201λ13 + ÿ(12) λ201λ23 + ÿ(12) λ01λ 2 03 + ÿ(6) λ203λ04 ) + 20 ¨ (ÿ(6) λ201λ02 + ÿ(6) λ203λ14 + ÿ(6) λ201λ34 ) + 60 ¨ (ÿ(6) λ01λ03λ13 + ÿ(1) λ03λ04λ34 ) + 110 ¨ (ÿ(6) λ01λ03λ04 + ÿ(6) λ01λ03λ23 + ÿ(6) λ01λ23λ14 + ÿ(2) λ01λ13λ14 ) + 120 ¨ (ÿ(12) λ01λ02λ13 + ÿ(12) λ01λ03λ14 + ÿ(12) λ01λ03λ24 + ÿ(12) λ01λ03λ34 + ÿ(6) λ01λ02λ03 ) + 200 ¨ (ÿ(3) λ01λ02λ34 ) , Σ1 = 10 ¨ (ÿ(12) λ201λ02λ03 + ÿ(12) λ201λ02λ13 + ÿ(12) λ201λ02λ23 + ÿ(12) λ201λ03λ14 + ÿ(12) λ203λ24λ34 + ÿ(12) λ01λ 2 03λ34 + ÿ(12) λ201λ03λ34 + ÿ(12) λ201λ03λ23 + ÿ(12) λ201λ03λ24 + ÿ(12) λ201λ13λ23 + ÿ(12) λ201λ13λ24 + ÿ(12) λ201λ13λ34 + ÿ(12) λ201λ23λ34 + ÿ(12) λ01λ 2 03λ04 + ÿ(12) λ01λ 2 03λ14 + ÿ(12) λ01λ 2 03λ24 + ÿ(6) λ201λ03λ04 + ÿ(6) λ201λ13λ14 + ÿ(6) λ201λ23λ24 + ÿ(6) λ01λ02λ 2 03 + ÿ(6) λ01λ 2 03λ23 ) + 20 ¨ (ÿ(6) λ201λ02λ34 + ÿ(3) λ203λ14λ24 ) + 40 ¨ (ÿ(12) λ01λ03λ04λ13 + ÿ(12) λ01λ02λ03λ13 + ÿ(6) λ01λ03λ04λ34 + ÿ(6) λ01λ03λ13λ14 + ÿ(6) λ01λ03λ13λ24 ) + 60 ¨ (ÿ(6) λ01λ03λ14λ34 ÿ(3) λ01λ02λ13λ23 ) + 80 ¨ (ÿ(12) λ01λ02λ03λ14 + ÿ(12) λ01λ03λ04λ23 + ÿ(12) λ01λ03λ14λ23 + ÿ(6) λ01λ02λ13λ14 + ÿ(6) λ01λ02λ13λ24 + ÿ(6) λ01λ03λ24λ34 + ÿ(3) λ01λ02λ03λ04 ) + 100 ¨ (ÿ(12) λ01λ02λ13λ34 + ÿ(6) λ01λ02λ03λ34 ) , Σ0 = 5 ¨ (ÿ(12) λ201λ02λ03λ14 + ÿ(12) λ201λ02λ03λ24 + ÿ(12) λ201λ02λ03λ34 + ÿ(12) λ201λ02λ13λ24 + ÿ(12) λ201λ02λ14λ23 + ÿ(12) λ201λ02λ23λ34 + ÿ(12) λ201λ03λ04λ23 + ÿ(12) λ201λ03λ14λ23 + ÿ(12) λ201λ03λ14λ24 + ÿ(12) λ201λ03λ23λ24 + ÿ(12) λ201λ03λ23λ34 + ÿ(12) λ201λ03λ24λ34 + ÿ(12) λ201λ13λ14λ23 + ÿ(12) λ201λ13λ23λ24 + ÿ(12) λ201λ13λ23λ34 + ÿ(12) λ201λ13λ24λ34 95 6. DV-CELLS AND NORMALISED SECOND MOMENTS + ÿ(12) λ01λ02λ 2 03λ14 + ÿ(12) λ01λ 2 03λ14λ24 + ÿ(12) λ01λ 2 03λ04λ23 + ÿ(12) λ01λ 2 03λ04λ24 + ÿ(12) λ01λ 2 03λ14λ23 + ÿ(12) λ01λ 2 03λ24λ34 + ÿ(6) λ201λ02λ03λ04 + ÿ(6) λ201λ02λ13λ14 + ÿ(6) λ201λ02λ23λ24 + ÿ(6) λ01λ02λ 2 03λ34 + ÿ(6) λ203λ04λ14λ24 ) + 20 ¨ (ÿ(12) λ01λ02λ03λ04λ13 + ÿ(12) λ01λ02λ03λ13λ14 + ÿ(12) λ01λ02λ03λ13λ24 + ÿ(12) λ01λ02λ03λ13λ34 + ÿ(12) λ01λ03λ04λ13λ24 + ÿ(6) λ01λ02λ13λ14λ34 + ÿ(6) λ01λ03λ04λ23λ34 + ÿ(3) λ01λ02λ03λ04λ34 ) + 30 ¨ (ÿ(12) λ01λ02λ03λ14λ34 + ÿ(6) λ01λ02λ03λ14λ24 + ÿ(6) λ01λ02λ13λ14λ23 + ÿ(6) λ01λ03λ04λ23λ24 ) + 40 ¨ (ÿ(6) λ01λ02λ13λ24λ34 ) . In addition we find for the determinant det(c) = ÿ4 i=0 λia ¨ Σdi, where Σd0 = ÿ(12) λ01λ02λ03λ14 + ÿ(12) λ01λ02λ13λ34 + ÿ(12) λ01λ03λ04λ23 + ÿ(12) λ01λ03λ14λ23 + ÿ(6) λ01λ02λ03λ34 + ÿ(6) λ01λ02λ13λ14 + ÿ(6) λ01λ02λ13λ24 + ÿ(6) λ01λ03λ24λ34 + ÿ(3) λ01λ02λ03λ04, Σd1 = 2 ¨ (ÿ(12) λ01λ02λ13 + ÿ(12) λ01λ03λ14 + ÿ(12) λ01λ03λ24 + ÿ(12) λ01λ03λ34 + ÿ(6) λ01λ02λ03 + ÿ(6) λ01λ03λ04 + ÿ(6) λ01λ03λ23 + ÿ(6) λ01λ13λ24 + ÿ(2) λ01λ13λ14 ) + 4 ¨ (ÿ(3) λ01λ02λ34 ) , Σd2 = 3 ¨ (ÿ(12) λ01λ03 + ÿ(6) λ01λ13 + ÿ(6) λ01λ23 + ÿ(3) λ03λ04 ) + 4 ¨ (ÿ(6) λ01λ34 + ÿ(3) λ01λ02 ) , Σd3 = 4 ¨ (ÿ(6) λ01 + ÿ(3) λ03 ) , Σd4 = 4. The local quantizer polynomial for ∆3: We continue with the rational closure ∆3, let D3 denote its associated Delone triangulation. Those maximal cells of the triangulation D3, that contain the origin 0, are given by the star of D3 around 0. Following [Val03, 8.4.2] we set D3I := ă e4, e3 ´ e4, e2 + e4, e1 ´ e2, e2 + e5 ą D3VII := ă e3, e1, e4, e2, e5 ą D3II := ă e4, e2 + e3, e1 ´ e2, e2, e5 ą D3VIII := ă e2 + e4, e3 ´ e4, e4, e1, e5 ą D3III := ă e2, e3, e1, e4, e5 ą D3IX := ă e2 + e4, e1 ´ e2, e2, e3, e5 ą D3IV := ă e1 + e4, e3 ´ e4, e4, e2, e5 ą D3X := ă e2 + e3, e1 ´ e2, e2, e4, e5 ą D3V := ă e1 ´ e2, e2, e4, e3 ´ e4, e2 + e4 + e5 ą D3XI := ă e1, e3, e2, e4, e5 ą D3VI := ă e2, e1 ´ e2, e2 + e4, e3, e5 ą D3III := ă e1, e4, e2, e3, e5 ą 96 6.4. ON THE LATTICE QUANTIZER PROBLEM IN LOW DIMENSIONS which follows Voronois numeration (cf. p. 173 in [Vor08]). Again we use the snake notation of Ryshkov (cf. (6.2)), where e5 = ´e1 ´ e2 ´ e3 ´ e4. IfD is any of the above simplices we denote its image under ´ id by the roman numeral that corresponds to 12 + iwhere i is the value of the roman numeral of D. D20 then consists of the translates of the above numbered 24 simplices by their vertices. This amounts to a total of 120 elements. Thus star(0,D3)(4) = tD31, . . . ,D3120u, where we numerate the elements of star(0,D3)(4) by starting with, say, D31 := D 3 I , followed by its translates D32, . . . ,D 3 5 and continuing with D 3 6 being D 3 II and so on. We compute Aut(∆3) to be the subgroup of S9, acting on the set t01, 02, 03, 04, 13, 14, 23, 24,bu of indices of the extreme rays, that is generated by σ1 = (01, 03, 23,b, 24, 04)(02, 14, 13), σ2 = (01, 23)(02, 13)(04,b), σ3 = (01, 04, 14)(02, 23, 03)(13, 24,b). This group has order 36 and is available as entry ă 36, 10 ą in the small group database via GAP or MAGMA (cf. [BE99]). In fact we compute the matrix group fixing Ea, which is isomorphic to Aut(D4) and afterwards the subgroup fixing ∆3. Aut(∆3) does unfortunately not act naturally on the conical parameters given the above explicit generators. Let us express our understanding that it is quite un- fortunate that we did not find a nicer description of this action, as was possible in the cases of ∆1 and ∆2. Doing this we can derive a “compressed” expression for G(c)∆2 , however it is unfortunately not as com- pressed as the one for G(c)∆1 , but fortunately nicer than the one for G (c) ∆2 . G (c) ∆3 = det(c) 240 ¨ ( 5 ¨ det(c) ¨ Σd + 5ÿ i=0 λia ¨ Σi ) , (6.12) where det(c) = ÿ4 i=0 λia ¨ Σdi, where Σ5 = 104, Σ4 = 130 ¨ (ÿ(9) λ01 ) , Σ3 = 20 ¨ (ÿ(9) λ201 ) + 130 ¨ (ÿ(18) λ01λ03 + ÿ(9) λ01λ23 ) + 160 ¨ (ÿ(9) λ01λ02 ) , Σ2 = 15 ¨ (ÿ(36) λ201λ03 + ÿ(18) λ201λ23 ) + 20 ¨ (ÿ(18) λ201λ02 ) + 60 ¨ (ÿ(6) λ01λ03λ13 ) + 110 ¨ (ÿ(18) λ01λ03λ04 + ÿ(3) λ01λ23λ24 ) + 120 ¨ (ÿ(36) λ01λ02λ13 + ÿ(18) λ01λ02λ03 ) + 200 ¨ (ÿ(3) λ01λ02λb ) , Σ1 = 10 ¨ (ÿ(36) λ201λ02λ03 + ÿ(36) λ201λ02λ14 + ÿ(36) λ201λ02λ23 + ÿ(36) λ201λ03λ23 + ÿ(36) λ201λ03λ24 + ÿ(18) λ201λ03λ04 + ÿ(18) λ201λ03λ14 + ÿ(9) λ201λ23λ24 ) + 20 ¨ (ÿ(9) λ201λ02λb ) 97 6. DV-CELLS AND NORMALISED SECOND MOMENTS + 40 ¨ (ÿ(36) λ01λ02λ03λ13 ) + 60 ¨ (ÿ(9) λ01λ02λ13λ23 ) + 80 ¨ (ÿ(18) λ01λ02λ03λ14 + ÿ(18) λ01λ02λ13λ14 + ÿ(18) λ01λ02λ13λ24 + ÿ(9) λ01λ02λ03λ04 ) + 100 ¨ (ÿ(18) λ01λ02λ03λb ) , Σ0 = 5 ¨ (ÿ(36) λ201λ02λ03λ14 + ÿ(36) λ201λ02λ03λ24 + ÿ(36) λ201λ02λ03λb + ÿ(36) λ201λ02λ13λ24 + ÿ(36) λ201λ03λ04λ23 + ÿ(36) λ201λ03λ23λ24 + ÿ(18) λ201λ02λ03λ04 + ÿ(18) λ201λ02λ13λ14 + ÿ(18) λ201λ02λ23λ24 + ÿ(18) λ201λ02λ23λb + ÿ(18) λ201λ03λ14λ23 + ÿ(18) λ201λ03λ14λ24 ) + 20 ¨ (ÿ(36) λ01λ02λ03λ04λ13 + ÿ(18) λ01λ02λ03λ13λ14 + ÿ(18) λ01λ02λ03λ13λb ) + 30 ¨ (ÿ(18) λ01λ02λ03λ14λ24 + ÿ(18) λ01λ02λ13λ14λ23 ) + 40 ¨ (ÿ(9) λ01λ02λ03λ04λb ) . In addition we find for the determinant det(c) = ÿ4 i=0 λia ¨ Σdi, where Σd0 = ÿ(18) λ01λ02λ03λ14 + ÿ(18) λ01λ02λ03λb + ÿ(18) λ01λ02λ13λ14 + ÿ(18) λ01λ02λ13λ24 + ÿ(9) λ01λ02λ03λ04, Σd1 = 2 ¨ (ÿ(36) λ01λ02λ13 + ÿ(18) λ01λ03λ02 + ÿ(18) λ01λ03λ04 + ÿ(3) λ01λ23λ24 ) + 4 ¨ (ÿ(3) λ01λ02λb ) , Σd2 = 3 ¨ (ÿ(18) λ01λ03 + ÿ(9) λ01λ23 ) + 4 ¨ (ÿ(9) λ01λ02 ) , Σd3 = 4 ¨ (ÿ(9) λ01 ) , Σd4 = 4. 6.4.4 Results on local optimality in dimension 4 We have already seen that there is only one local, and therefore global, optimum of the quantizer problem in dimensions 2 and 3. For dimensions larger than 3 we can take a look on certain well-known lattices and check them for local optimality for the lattice quantizer problem, by use of the explicit formula for the quantizer constant given in Corollary 6.3.9. Given a lattice L we compute a coordinate representation (Zn,Q). We can then compute ∆(DelZn(Q)) and then all secondary cones of triangulations that refine DelZn(Q). For 98 6.4. ON THE LATTICE QUANTIZER PROBLEM IN LOW DIMENSIONS each secondary cone of a triangulation we then compute the explicit term for G(c). In fact, since the conic parameters are homogeneous it is possible to assume that one of them is equal to 1, without any loss of generality, leading to a term G 1(c). Next we compute the gradient and hessian of the altered explicit term G 1(c) using the symbolic engine of MAPLE. We do so for dimension 4. THEOREM 6.4.3. The lattice D4 – D#4 is a local minimum for the lattice quantizer problem in Dimension 4. PROOF. For D4 we have to consider ∆2 and ∆3, the lattice corresponds to the extreme ray Ea which both cones share. In ∆2 we assume that G 1(c) comes from λa = 1 and find D4 to be represented by [0, 0, 0, 0, 0, 0, 0, 0, 0, 1] and∇G 1(c) vanishes at the truncated point. In∆3we assume thatG 1(c) comes from λa = 1 and find D4 to be represented by [0, 0, 0, 0, 0, 0, 0, 0, 1, 0] and ∇G 1(c) vanishes at the truncated point. The hessian, evaluated at this point, is 43/4 1536 ¨  3 ´1 0 0 0 0 0 0 ´1 ´1 3 0 0 0 0 0 0 ´1 0 0 3 0 0 ´1 0 ´1 0 0 0 0 3 ´1 0 ´1 0 0 0 0 0 ´1 3 0 ´1 0 0 0 0 ´1 0 0 3 0 ´1 0 0 0 0 ´1 ´1 0 3 0 0 0 0 ´1 0 0 ´1 0 3 0 ´1 ´1 0 0 0 0 0 0 3  regardless of the cone. This matrix is positive definite, where the integral part has eigenvalues 1 and 4 of multiplicities 3 and 6 respectively. THEOREM 6.4.4. The lattice A#4 is a local minimum for the lattice quantizer problem in Dimension 4. PROOF. For A#4 we only have to consider ∆1. We assume that G 1 (c) comes from λ01 = 1. A # 4 is repre- sented by [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] and ∇G 1(c) vanishes at the truncated point. The hessian, evaluated at this point, is 51/4 150000 ¨  111 14 14 14 ´65 ´65 14 14 ´65 14 111 14 ´65 14 ´65 14 ´65 14 14 14 111 ´65 ´65 14 ´65 14 14 14 ´65 ´65 111 14 14 14 14 ´65 ´65 14 ´65 14 111 14 14 ´65 14 ´65 ´65 14 14 14 111 ´65 14 14 14 14 ´65 14 14 ´65 111 14 14 14 ´65 14 14 ´65 14 14 111 14 ´65 14 14 ´65 14 14 14 14 111  . This matrix is positive definite, where the integral part has eigenvalues 81 + 3 ? 678, 81 + 3 ? 678, 255, and 18 of multiplicities 1, 1, 3, and 4 respectively. THEOREM 6.4.5. The lattice A4 is not a local minimum of the lattice quantizer problem in Dimension 4, it is, however, a saddle point. 99 6. DV-CELLS AND NORMALISED SECOND MOMENTS PROOF. For A4 we only have to consider ∆1. We assume that G 1(c) comes from λ01 = 1. A4 is repre- sented by [1, 0, 0, 1, 1, 0, 0, 1, 0, 1] and ∇G 1(c) vanishes at the truncated point. The hessian, evaluated at this point, is 53/4 6000 ¨  ´3 ´3 ´2 3 2 2 ´2 ´3 ´2 ´3 ´3 3 ´2 ´3 2 ´2 2 3 ´2 3 12 ´3 ´2 3 ´3 ´2 ´3 3 ´2 ´3 12 3 ´2 ´3 ´2 ´3 2 ´3 ´2 3 ´3 ´3 3 2 ´2 2 2 3 ´2 ´3 ´3 ´2 ´3 ´2 ´2 ´2 ´3 ´3 3 ´2 12 3 ´3 ´3 2 ´2 ´2 2 ´3 3 ´3 3 ´2 3 ´3 ´3 ´2 ´2 ´3 3 12  . This matrix is regular, but indefinite. 100 APPENDIX A Table of quantizer constants We present an overview of some explicit values of the quantizer constant for low dimensions. The table contains the currently known best monohedral periodic quantizers in dimensions 1 to 10 as well as those in dimensions 12 and 16, which are lattices with the exception of dimension 7 and 9. The explicit values for A#3 – D#3 appears on p.378 in [Ger79]. Explicit formulae for the quantizer constant of the root lattice families An,A#n (for n ě 1) and Dn,D#n (for n ě 3) as well as the explicit values for the exceptional lattices E6,E7,E8 can be found in Chapter 21 of [CS98], cf. [CS82]. The approximate value of BW16 is from 2.3, Chapter 2 in [CS98], cf. [CS84]. The exact formulae for E#6,E # 7 can be found in [Wor87], [Wor88]. The approximate values for D+7 ,D + 9 are from Table IV [AE98]. The explicit values for D+10 and K12 are from Table 5 in [DSSV09].. Dimension Quantizer Normalized second moment source 1 Z 112 « 0.83333 [CS98] 2 A#2 – A2 536 ¨ 3´1/2 « 0.080188 [CS98] 3 A#3 – D#3 19192 ¨ 2´1/3 « 0.078543 [CS98] A3 – D3 2´11/3 « 0.078745 [CS98] 4 D#4 – D4 13120 ¨ 2´1/2 « 0.076603 [CS98] A#4 389 1500 ¨ 5´3/4 « 0.077559 [CS98] A4 7 60 ¨ 5´1/4 « 0.078020 [CS98] 5 D#5 2641 23040 ¨ 2´3/5 « 0.075625 [CS98] D5 1 10 ¨ 2´2/5 « 0.075786 [CS98] A#5 209 648 ¨ 6´4/5 « 0.076922 [CS98] A5 1 9 ¨ 6´1/5 « 0.077647 [CS98] 101 A. TABLE OF QUANTIZER CONSTANTS Dimension Quantizer Normalized second moment source 6 E#6 12619 204129 ¨ 31/6 « 0.074244 [Wor87] E6 5 56 ¨ 3´1/60.074347 [CS98] 7 D+7 0.072734 ˘ 0.000003 [AE98] E#7 21361 322560 ¨ 21/70.073116 [Wor88] E7 163 2016 ¨ 2´1/7 « 0.073231 [CS98] 8 E#8 – E8 92912960 « 0.071682 [CS98] 9 D+9 0.071103 ˘ 0.000003 [AE98] 10 D+10 4568341 64512000 « 0.070813 [DSSV09] 12 K12 7973619416567561000 ¨ 3´1/2 « 0.070095 [DSSV09] 16 Λ16 – BW16 0.068299 ˘ 0.000027 [CS98] 102 APPENDIX B Documentation of computations B.1 METHOD OF COMPUTATION To find the explicit expression for the quantizer constant, as proposed by Corollary 6.3.9, in the cases depicted in 6.4, we used the following approach involving the computer algebra systems MAGMA [BCP97] and MAPLE [Map]. Let ∆(D) be the secondary cone of the Delone subdivision D for which we are interested in an explicit formula. i. Choose a labeling I for the full dimensional Delone polytopes in star(D, 0). ii. Choose Q0 P ∆ and compute the vertices of its Dirichlet-Voronoi cell in order of the labeling I. iii. In MAGMA: The intrinsic command Polytope computes the Dirichlet-Voronoi cell of Q0 in terms of the labeling I, the intrinsic command Faces computes the full face lattice FL. iv. In MAPLE: Convert FL into suitable input FaceLatt for the command face_to_polytope (cf. B.2). v. In MAPLE: For all elements F of FaceLatt compute the pulling triangulation (in the inher- ent vertex order of the labeling) of face_to_polytope(FaceLatt,F), using the command pulling_triangulation (cf. B.2). vi. In MAPLE: The command quantizerformula (cf. B.2) computes the sought after expression. We quickly explain what a suitable input for the command pulling_triangulation looks like. An n-polytope P, which we know in terms of its face lattice F, is converted in the following way: for 0 ď k ă n, a k-face of P, given in terms of the labels of the vertices as [i1, . . . , ij], is converted to [k, ti1, . . . , iku], that is, for each face of P we store its dimension and its vertices. This produces a representation F. The command face_to_polytope will take F and an element F P F as input and compute an anal- ogously representation of F, given in terms of its faces [k 1, tij1 , . . . , ijl1 u]. 103 B. DOCUMENTATION OF COMPUTATIONS B.2 OVERVIEW OF ROUTINES centroid(S,Q) Returns the centroid of the simplex with vertices in the list S with respect to the quadratic form given by a matrix Q. q_centroid(S,Q) Returns Q times the centroid of the simplex with vertices in the list S with respect to the quadratic form given by a matrix Q. simplexvolume(S) Returns the volume of the simplex with vertices in the list S with respect to the standard inner product. adj_norm(v,G) Returns the squared norm of a vector v (can also be the list of coefficients) with respect to the adjoint of the matrix G. simplexsummand(S,Q,S_0) Returns the quantizer summand (cf. Corollary 6.3.4) belonging to the simplex with ver- tices in the list S cat [0]. S_0 has to be a list of vertices of an explicit realization of S in the same order as S, it is used to determine the sign of the orientation (cf. 6.3.6). quantizerformula(Del,VorTriangulation,Q) S - list; VorTriangulation - list; Q - matrix; Returns the quotient for the quantizer con- stant. The input has to be: Del - a list of the full dimensional Delone polytopes containing 0; VorTriangulation - a list containing an abstract triangulation of the Dirichlet-Voronoi cell; Q - a matrix representing an element of the interior of the associated secondary cone. pyramid(P,v) Returns the pyramid of v (given as a list) over P. vertices(P) Returns all vertices of P. face_to_polytope(P.F) Returns the face F of P as a polytope. 104 B.3. WORKSHEETS pull(L,v) Returns the pulling refinement of L by pulling v (cf. 4.1). pulling_triangulation(P) Returns the pulling triangulation of P corresponding to the vertex order in which P is given (cf. Lemma 4.2.1). SJT_alg(n) Returns the symmetric group on n symbols in the order produced by the Steinhaus- Johnson-Trotter algorithm. use_sym_binary(sigma,M) Returns the result of the operation of sigma on the binary symbols in M. use_sym_unary(sigma,M) Returns the result of the operation of sigma on the unary symbols in M. sym_orbit_sum_red_binary(M,Sym) Returns the polynomial in l that is the sum of the monomials indexed by sigma(M) for sigma in Sym, M a list of binary indices. Multiplicities are removed. sym_orbit_sum_red_unary(M,Sym) Returns the polynomial in l that is the sum of the monomials indexed by sigma(M) for sigma in Sym, M a list of unary indices. Multiplicities are removed. expand_snake(S) Returns a list of the vertices of the Simplex described by vertices in the list S, which is interpreted to be in snake notation (cf. 6.2). B.3 WORKSHEETS The routines presented in B.2 are available online at https://marcchristianzimmermann.wordpress.com/ The computations for dimension 4 (cf. 6.4.3, 6.4.4) are also available at the above address: for each of the three Delone triangulations we provide a text file containing a MAPLE script intended to be a check and documentation of the involved computations. 105 Index L-fundamental domain, 53 b-primitive, 4 anisotropic element, 5 module, 5 associated bilinear form, 2 automorphism group of a lattice, 7 of a periodic set, 53 basis matrix lattice, 52 bilinear module, 2 complex boundary, 56 polytopal, 56 conic parameters, 84 Delone set, 59 isohedral, 60 monohedral, 60 symmetric, 59 determinant of a bilinear module, 4 of a quadratic module, 4 determinant-index formula, 7 Dirichlet-Voronoi cell of a discrete set, 60 Dirichlet-Voronoi subdivision, 60 dual module, 4 fundamental parallelotope, 52 genus, 8 Gram matrix, 4 lattice, 7 Hasse-symbol, 6 Hilbert-symbol, 6 hyperbolic module, 5 plane, 5 hyperbolic pair, 5 isometric bilinear modules, 3 quadratic modules, 3 isometry of a bilinear module, 3 of a lattice, 7 of a quadratic module, 3 isometry group of a bilinear module, 4 of a lattice, 7 of a quadratic module, 3 isotropic element, 5 module, 5 lattice, 6 basis, 6 coordinate representation, 54 dual, 7 basis, 8 107 INDEX partial, 8 Euclidean, 53 Gram matrix, 6 isometry class, 7 maximal, 7 on, 6 real space, 52 standard, 52 sublattice, 6 unigeneric, 8 unimodular, 8 volume, 53 localization of Q, 5 of a lattice, 8 of a rationals space, 8 of a ring of integers, 5 orthogonal group of a lattice, 7 of a periodic set, 53 orthogonal sum internal, 4 orthogonality of elements, 4 periodic form, 55 periodic set, 52 t-representable, 55 coordinate representation, 55 isohedral, see Delone set monohedral, see Delone set standard, 53 translation lattice, 52 volume, 62 polytope face lattice, 56 primitive, 4 pulling refinement, 56 quadratic form, 2 quadratic module, 2 quadratic space, see quadratic module quantizer constant, 66 regular bilinear form, 4 quadratic form, 5 scaling, 7 second moment, 65 snake notation, 85 star, 56 subdivision polytopal, 56 trivial, 56 tiling, 59 isohedral, 59 monohedral, 59 totally isotropic module, 5 triangulation, 56 pulling, 57 pyramidal, 56 vector quantizer, 66 lattice, 68 periodic monohedral, 68 108 Bibliography [AE98] E. 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