Maximal periods of (Ehrhart) quasi-polynomials

Title:
Maximal periods of (Ehrhart) quasi-polynomials
Authors:
Woods, Kevin; Beck, Matthias; Sam, Steven V.
Abstract:
A quasi-polynomial is a function defined of the form q(k)=cd(k)kd+cd−1(k)kd−1+⋯+c0(k)q(k)=cd(k)kd+cd−1(k)kd−1+⋯+c0(k), where c0,c1,…,cdc0,c1,…,cd are periodic functions in k∈Zk∈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 cj(k)cj(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.
Citation:
Woods, Kevin, Matthias Beck, and Steven Sam. 2008. "Maximal periods of (Ehrhart) quasi-polynomials." Journal Of Combinatorial Theory 115(3): 517-525.
Publisher:
Elsevier Science
DATE ISSUED:
2008
Department:
Mathematics
Type:
article
PUBLISHED VERSION:
10.1016/j.jcta.2007.05.009
PERMANENT LINK:
http://hdl.handle.net/11282/309636

Full metadata record

DC FieldValue Language
dc.contributor.authorWoods, Kevinen_US
dc.contributor.authorBeck, Matthiasen_US
dc.contributor.authorSam, Steven V.en_US
dc.date.accessioned2013-12-23T16:13:45Zen
dc.date.available2013-12-23T16:13:45Zen
dc.date.issued2008en
dc.identifier.citationWoods, Kevin, Matthias Beck, and Steven Sam. 2008. "Maximal periods of (Ehrhart) quasi-polynomials." Journal Of Combinatorial Theory 115(3): 517-525.en_US
dc.identifier.issn0097-3165en_US
dc.identifier.urihttp://hdl.handle.net/11282/309636en
dc.description.abstractA quasi-polynomial is a function defined of the form q(k)=cd(k)kd+cd−1(k)kd−1+⋯+c0(k)q(k)=cd(k)kd+cd−1(k)kd−1+⋯+c0(k), where c0,c1,…,cdc0,c1,…,cd are periodic functions in k∈Zk∈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 cj(k)cj(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.en_US
dc.publisherElsevier Scienceen_US
dc.identifier.doi10.1016/j.jcta.2007.05.009en
dc.subject.departmentMathematicsen_US
dc.titleMaximal periods of (Ehrhart) quasi-polynomialsen_US
dc.typearticleen_US
dc.identifier.journalJournal Of Combinatorial Theoryen_US
dc.identifier.volume115en_US
dc.identifier.startpage517en_US
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