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. |
Contents
Notes | 48 |
Open Problems | 54 |
Notes | 83 |
Exercises | 90 |
Exercises | 103 |
Finite Fourier Analysis | 123 |
Dedekind Sums the Building Blocks of Latticepoint | 138 |
The Decomposition of a Polytope into Its Cones | 155 |
EulerMaclaurin Summation in Rd 167 | 166 |
Solid Angles | 179 |
A Discrete Version of Greens Theorem Using Elliptic | 191 |
A Vertex and Hyperplane Descriptions of Polytopes | 199 |
Hints for Exercises | 209 |
References | 217 |
List of Symbols | 227 |
Other editions - View all
Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra Matthias Beck,Sinai Robins No preview available - 2010 |
Common terms and phrases
Algebra algorithm Ap(t Bernoulli polynomials Brion's theorem Chapter coefficients combinatorics compute cone(P const constant term continuous volume convex polytope Corollary counting function cross-polytope d-cone d-polytope Dedekind sums defined denote dimension discrete volume Ehrhart polynomial Ehrhart quasipolynomial Ehrhart series Ehrhart's theorem Ehrp example Exercise face finite Fourier series 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 Number Theory PA(n partial fraction expansion Pick's theorem pointed cone polygon prime positive integers proof of Theorem Prove rational convex rational function rational polytopes reciprocity law relatively prime relatively prime positive restricted partition function right-hand side roots of unity Show simplex simplices simplicial cones solid angle Suppose triangulation v a vertex vector vertex cone vertices Zd+1
References to this book
Integer Points in Polyhedra-- Geometry, Number Theory, Representation Theory ... Matthias Beck No preview available - 2008 |