Maximal periods of (Ehrhart) quasi-polynomials

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...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series A 2008-04, Vol.115 (3), p.517-525
Main Authors: Beck, Matthias, Sam, Steven V., Woods, Kevin M.
Format: Article
Language:English
Subjects:
Ehrhart quasi-polynomial
Lattice points
Period
Quasi-polynomial convolution
Rational polytope
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Summary: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.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2007.05.009