A finite calculus approach to Ehrhart polynomials

TL;DR

This paper introduces a novel proof of Ehrhart's theorem using finite calculus, providing new insights into the properties of Ehrhart quasi-polynomials and their coefficients.

Contribution

It presents a new finite calculus-based proof of Ehrhart's theorem and related properties like coefficient periodicity and reciprocity.

Findings

New proof of Ehrhart's theorem using finite calculus

Proof of McMullen's periodicity theorem for coefficients

Verification of Ehrhart-Macdonald reciprocity

Abstract

A rational polytope is the convex hull of a finite set of points in $\R^d$ with rational coordinates. Given a rational polytope $P \subseteq \R^d$, Ehrhart proved that, for $t\in\Z_{\ge 0}$, 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.