A finite calculus approach to Ehrhart polynomials

Abstract

A rational polytope is the convex hull of a finite set of points in ${\mathbb{R}}^d$ with rational coordinates. Given a rational polytope $P\subseteq {\mathbb{R}}^d$, Ehrhart proved that, for $t\in {\mathbb{Z}}_{\ge 0}$, the function $#(tP \cap \Z^d)$ agrees with a quasi-polynomial $L_P(t)$, called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart's theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen's theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart-Macdonald theorem on reciprocity.

13 pages, 1 figure; v2: added examples and Section 4, final version

References in corpus (2)

• Counting with rational generating functions
• Maximal Periods of (Ehrhart) Quasi-Polynomials