A Finite Calculus Approach to Ehrhart Polynomials
Author(s)
Sam, Steven V.; Woods, Kevin M.
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A rational polytope is the convex hull of a finite set of points in R[superscript d] with rational coordinates. Given a rational polytope P⊆R[superscript d], Ehrhart proved that, for t∈Z≥[subscript 0[, the function #(tP∩Z[superscript d]) agrees with a quasi-polynomial L[subscript 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.
Date issued
2010-04Department
Massachusetts Institute of Technology. Department of MathematicsJournal
Electronic Journal of Combinatorics
Publisher
Electronic Journal of Combinatorics
Citation
Sam, Steven V., and Kevin M. Woods. "A Finite Calculus Approach to Ehrhart Polynomials." Electronic Journal of Combinatorics, Volume 17 (2010).
Version: Final published version
ISSN
1077-8926