Maximal Periods of (Ehrhart) Quasi-Polynomials

Abstract

A \emph{quasi-polynomial} is a function defined of the form $q(k)=c_d(k)k^d+c_{d-1}(k)k^{d-1}+...+c_0(k)$, where $c_0,c_1,...,c_d$ are periodic functions in $k\in \mathbb{Z}$. Prominent examples of quasi-polynomials appear in Ehrhart's theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the $c_j(k)$ for Ehrhart quasi-polynomials. For generic polytopes, McMullen's bounds seem to be sharp, but sometimes smaller periods exist. We prove that the second leading coefficient of an Ehrhart quasi-polynomial always has maximal expected period and present a general theorem that yields maximal periods for the coefficients of certain quasi-polynomials. We present a construction for (Ehrhart) quasi-polynomials that exhibit maximal period behavior and use it to answer a question of Zaslavsky on convolutions of quasi-polynomials.

7 pages, to appear in JCT-A