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 .
