TAILIEUCHUNG - Báo cáo toán học: "A finite calculus approach to Ehrhart polynomials"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A finite calculus approach to Ehrhart polynomials. | A finite calculus approach to Ehrhart polynomials Steven V. Sam Department of Mathematics Massachusetts Institute of Technology ssam@ http ssam Kevin M. Woods Department of Mathematics Oberlin College http f aculty kwoods Submitted Nov 24 2009 Accepted Apr 20 2010 Published Apr 30 2010 Mathematics Subject Classification 52C07 Abstract A rational polytope is the convex hull of a finite set of points in Rd with rational coordinates. Given a rational polytope P c Rd Ehrhart proved that for t G Z o the function tP n Zd agrees with a quasi-polynomial Lp 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. 1 Introduction. Let us first look at an easy example of computing a volume. Let Ad c Rd be the convex hull of the following d 1 points the origin and the standard basis vectors Ci 1 i d. Let tAd be its dilation by a factor of t for nonnegative t . A straightforward way of computing the volume of tAd would be inductively in d using the fact that the d 1 -dimensional cross section of tAd at xd s is a dilated copy of Ad-1 vol tAd Ị vol sAd-1 ds and evaluating this iteratively gives us vol tAd td d . A generalization of volume is the Ehrhart quasi- polynomial which we define as follows. A polytope P is the convex hull of finitely many points in Rn and the dimension dim P of the polytope is the dimension of the affine hull of P. A rational resp. integral polytope is a polytope all of whose vertices are rational resp. integral . Given a THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R68 1 polytope P and a nonnegative t let tP be the .

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