A Finite Calculus Approach to Ehrhart Polynomials
Name
Sam-2010-Finite calculus approach.pdf
Size
139.86 KB
Format
Adobe PDF
Checksum (MD5)
a60a8808d798f41b4b16dba2c2aa9346
Author(s)
Sam, Steven V.
Woods, Kevin M.
Date Issued
April 2010
Journal
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
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.37236/340
