## Computing the Continuous Discretely: Integer-point Enumeration in PolyhedraThe 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 |

229 | |

### Common terms and phrases

Algebra algorithm Bernoulli polynomials Brion’s theorem Chapter coeﬃcients Coin-Exchange Problem compute cone(P const constant term continuous volume convex polytope Corollary counting function cross-polytope d-cone d-polytope Dedekind sums deﬁned deﬁnition denote dimension discrete volume Ehrhart polynomial Ehrhart quasipolynomial Ehrhart series Ehrhart–Macdonald reciprocity Ehrhart’s theorem example Exercise face ﬁnite Fourier finite Fourier series ﬁrst formula Fourier series Fourier–Dedekind sums Frobenius problem geometry identity integer points integer-point transform integral polytope lattice points lattice-point enumerator Lemma line segment linear LP(t magic squares Math Mathematics matrix Open Problems pA(n partial fraction expansion Pick’s theorem pointed cone polygon prime positive integers proof of Theorem Prove rational convex rational function rational polytopes rational triangle reciprocity law relatively prime relatively prime positive right-hand side roots of unity Second edition sequence Show simplex simplices simplicial cones solid angle spanF Suppose triangulation v a vertex vector vertex cone vertices