Note: Maximal periods of (Ehrhart) quasi-polynomials

Authors:
Matthias Beck;Steven V. Sam;Kevin M. Woods
Affiliations:
Department of Mathematics, San Francisco State University, San Francisco, CA 94132, USA;Department of Mathematics, University of California, Berkeley, CA 94720, USA;Department of Mathematics, Oberlin College, Oberlin, OH 44074, USA
Venue:
Journal of Combinatorial Theory Series A
Year:
2008

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Abstract

A 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@?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.