Maximal Periods of (Ehrhart) Quasi-Polynomials

Details Maximal Periods of (Ehrhart) Quasi-Polynomials Maximal Periods of (Ehrhart) Quasi-Polynomials

2007-02-09

arxiv.org/abs/math/0702242v3

J. Combin. Theory Ser. A 115, no. 3 (2008), 517-525

Mathematics

Combinatorics

7 pages, to appear in JCT-A

Scientific paper

10.1016/j.jcta.2007.05.009

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 $k \in \Z$. 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.

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