A Finite Calculus Approach to Ehrhart Polynomials

Abstract

A rational polytope is the convex hull of a finite set of points in ${\Bbb R}^d$ with rational coordinates. Given a rational polytope ${\cal P} \subseteq {\Bbb R}^d$, Ehrhart proved that, for $t\in{\Bbb Z}_{\ge 0}$, the function $\#(t{\cal P} \cap {\Bbb Z}^d)$ agrees with a quasi-polynomial $L_{\cal P}(t)$, 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.