A Finite Calculus Approach To Ehrhart Polynomials

Title:
A Finite Calculus Approach To Ehrhart Polynomials
Authors:
Sam, Steven V.; Woods, Kevin
Abstract:
A rational polytope is the convex hull of a finite set of points in R-d with rational coordinates. Given a rational polytope P subset of R-d, Ehrhart proved that, for t is an element of Z(>= 0), the function #(tP boolean AND Z(d)) agrees with a quasi-polynomial L-P(t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart's theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen's theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart Macdonald theorem on reciprocity.
Citation:
Sam, Steven V., and Kevin M. Woods. 2010. "A Finite Calculus Approach To Ehrhart Polynomials." Electronic Journal Of Combinatorics 17(1): 68-R68.
Publisher:
Electronic Journal Of Combinatorics
DATE ISSUED:
2010-04
Department:
Mathematics
Type:
article
Additional Links:
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r68
PERMANENT LINK:
http://hdl.handle.net/11282/309921

Full metadata record

DC FieldValue Language
dc.contributor.authorSam, Steven V.en_US
dc.contributor.authorWoods, Kevinen_US
dc.date.accessioned2013-12-23T16:21:04Z-
dc.date.available2013-12-23T16:21:04Z-
dc.date.issued2010-04en
dc.identifier.citationSam, Steven V., and Kevin M. Woods. 2010. "A Finite Calculus Approach To Ehrhart Polynomials." Electronic Journal Of Combinatorics 17(1): 68-R68.en_US
dc.identifier.issn1077-8926en_US
dc.identifier.urihttp://hdl.handle.net/11282/309921-
dc.description.abstractA rational polytope is the convex hull of a finite set of points in R-d with rational coordinates. Given a rational polytope P subset of R-d, Ehrhart proved that, for t is an element of Z(>= 0), the function #(tP boolean AND Z(d)) agrees with a quasi-polynomial L-P(t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart's theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen's theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart Macdonald theorem on reciprocity.en_US
dc.language.isoen_USen_US
dc.publisherElectronic Journal Of Combinatoricsen_US
dc.relation.urlhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r68en_GB
dc.subject.departmentMathematicsen_US
dc.titleA Finite Calculus Approach To Ehrhart Polynomialsen_US
dc.typearticleen_US
dc.identifier.journalElectronic Journal Of Combinatoricsen_US
dc.subject.keywordMathematics, applieden_US
dc.subject.keywordMathematicsen_US
dc.identifier.volume17en_US
dc.identifier.issue1en_US
dc.identifier.startpage68en_US
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