# 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.

## Repository Citation

Sam, Steven V., and Kevin M. Woods. 2010. "A Finite Calculus Approach To Ehrhart Polynomials." Electronic Journal Of Combinatorics 17(1): 68-R68.

## 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