Periods of Ehrhart coefficients of rational polytopes

TL;DR

This paper investigates the periodic behavior of Ehrhart quasi-polynomials associated with rational polytopes, constructing examples with specific prescribed periods for their coefficient functions.

Contribution

It provides new constructions of rational polytopes whose Ehrhart quasi-polynomials have coefficient functions with predetermined periods.

Findings

Constructed families of polytopes with specified coefficient periods

Advances understanding of the possible period structures in Ehrhart quasi-polynomials

Progress towards characterizing which quasi-polynomials are Ehrhart polynomials

Abstract

Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a quasi-polynomial function of $k$ -- that is, a "polynomial" in which the coefficients are themselves periodic functions of $k$. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values.