A bijective proof for a theorem of Ehrhart

TL;DR

This paper presents a new geometric bijective proof for Ehrhart's theorem on the quasi-polynomial nature of counting integer points in dilated rational polytopes, also establishing Ehrhart reciprocity.

Contribution

It introduces a novel bijective approach to prove Ehrhart's theorem and reciprocity, enhancing understanding of the combinatorial structure involved.

Findings

Established a geometric bijection proof for Ehrhart's theorem

Proved Ehrhart reciprocity using the new methods

Demonstrated the quasi-polynomiality of the counting function

Abstract

We give a new proof for a theorem of Ehrhart regarding the quasi-polynomiality of the function that counts the number of integer points in the integral dilates of a rational polytope. The proof involves a geometric bijection, inclusion-exclusion, and recurrence relations, and we also prove Ehrhart reciprocity using these methods.