# Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra

Springer Science & Business Media, Nov 19, 2007 - Mathematics - 227 pages
The world is continuous, but the mind is discrete. David Mumford We seek to bridge some critical gaps between various ?elds of mathematics by studying the interplay between the continuous volume and the discrete v- ume of polytopes. Examples of polytopes in three dimensions include crystals, boxes, tetrahedra, and any convex object whose faces are all ?at. It is amusing to see how many problems in combinatorics, number theory, and many other mathematical areas can be recast in the language of polytopes that exist in some Euclidean space. Conversely, the versatile structure of polytopes gives us number-theoretic and combinatorial information that ?ows naturally from their geometry. Fig. 0. 1. Continuous and discrete volume. The discrete volume of a body P can be described intuitively as the number of grid points that lie inside P, given a ?xed grid in Euclidean space. The continuous volume of P has the usual intuitive meaning of volume that we attach to everyday objects we see in the real world. VIII Preface Indeed, the di?erence between the two realizations of volume can be thought of in physical terms as follows. On the one hand, the quant- level grid imposed by the molecular structure of reality gives us a discrete notion of space and hence discrete volume. On the other hand, the N- tonian notion of continuous space gives us the continuous volume.

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### Contents

 The CoinExchange Problem of Frobenius 3 12 Two Coins 5 13 Partial Fractions and a Surprising Formula 7 14 Sylvesters Result 11 15 Three and More Coins 13 Notes 15 Exercises 17 Open Problems 23
 Exercises 119 Open Problems 120 Finite Fourier Analysis 123 72 Finite Fourier Series for Periodic Functions on Z 125 73 The Finite Fourier Transform and Its Properties 129 74 The Parseval Identity 131 75 The Convolution of Finite Fourier Series 133 Notes 135

 A Gallery of Discrete Volumes 25 22 The Unit Cube 26 23 The Standard Simplex 29 24 The Bernoulli Polynomials as LatticePoint Enumerators of Pyramids 31 25 The LatticePoint Enumerators of the CrossPolytopes 36 26 Picks Theorem 38 27 Polygons with Rational Vertices 41 28 Eulers Generating Function for General Rational Polytopes 45 Notes 48 Exercises 50 Open Problems 54 Counting Lattice Points in Polytopes The Ehrhart Theory 57 32 IntegerPoint Transforms for Rational Cones 60 33 Expanding and Counting Using Ehrharts Original Approach 64 34 The Ehrhart Series of an Integral Polytope 67 35 From the Discrete to the Continuous Volume of a Polytope 71 36 Interpolation 73 37 Rational Polytopes and Ehrhart Quasipolynomials 75 38 Reﬂections on the CoinExchange Problem and the Gallery of Chapter 2 76 Exercises 77 Open Problems 82 Reciprocity 83 41 Generating Functions for Somewhat Irrational Cones 84 42 Stanleys Reciprocity Theorem for Rational Cones 86 43 EhrhartMacdonald Reciprocity for Rational Polytopes 87 44 The Ehrhart Series of Reﬂexive Polytopes 88 45 More Reflections on the CoinExchange Problem and the Gallery of Chapter 2 90 Exercises 91 Open Problems 93 Face Numbers and the DehnSommerville Relations in Ehrhartian Terms 94 52 DehnSommerville Extended 97 53 Applications to the Coefficients of an Ehrhart Polynomial 98 54 Relative Volume 100 Notes 102 Exercises 103 Magic Squares 105 61 Its a Kind of Magic 106 Integer Points in the Birkhoffvon Neumann Polytope 108 63 Magic Generating Functions and ConstantTerm Identities 111 64 The Enumeration of Magic Squares 116 Notes 117
 Dedekind Sums the Building Blocks of Latticepoint Enumeration 138 82 The Dedekind Sum and Its Reciprocity and Computational Complexity 143 83 Rademacher Reciprocity for the FourierDedekind Sum 144 84 The MordellPommersheim Tetrahedron 147 Notes 150 Exercises 151 Open Problems 153 The Decomposition of a Polytope into Its Cones 155 92 Tangent Cones and Their Rational Generating Functions 159 93 Brions Theorem 160 94 Brion Implies Ehrhart 162 Notes 163 Exercises 164 EulerMaclaurin Summation Rd 166 102 A Continuous Version of Brions Theorem 170 103 Polytopes Have Their Moments 172 104 From the Continuous to the Discrete Volume of a Polytope 174 Notes 176 Exercises 177 Open Problems 178 Solid Angles 179 112 SolidAngle Generating Functions and a BrionType Theorem 182 113 SolidAngle Reciprocity and the BrianchonGram Relations 184 114 The Generating Function of Macdonalds SolidAngle Polynomials 188 Notes 189 Open Problems 190 A Discrete Version of Greens Theorem Using Elliptic Functions 191 122 The Weierstraß and ζ Functions 193 123 A ContourIntegral Extension of Picks Theorem 195 Notes 196 Open Problems 197 Vertex and Hyperplane Descriptions of Polytopes 199 A1 Every hcone is a vcone 200 A2 Every vcone is an hcone 202 Triangulations of Polytopes 204 Hints for Exercises 209 References 217 List of Symbols 227 Index 229 Copyright