A Finite Calculus Approach To Ehrhart Polynomials

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 subset of R-d, Ehrhart proved that, for t is an element of Z(>= 0), the function #(tP boolean AND 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.

Publisher

Electronic Journal Of Combinatorics

Publication Date

4-1-2010

Publication Title

Electronic Journal Of Combinatorics

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.37236/340

Keywords

Mathematics, applied, Mathematics

Language

English

Format

text

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