A finite calculus approach to Ehrhart polynomials

作者：Sam Steven V^*; Woods Kevin M

来源：Electronic Journal of Combinatorics, 2010, 17(1): R68.

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.

• 出版日期2010-4-30