Periods of Ehrhart coefficients of rational polytopes

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.

10 pages

Cited by in corpus (1)

• On the period collapse of a family of Ehrhart quasi-polynomials