Lattice invariant valuations on rational polytopes

Abstract

LetΛ be a lattice ind-dimensional euclidean space$\mathbb{E}^d$, and$\bar \Lambda$ the rational vector space it generates. Ifϕ is a valuation invariant underΛ, andP is a polytope with vertices in$\bar \Lambda$, then for non-negative integersn there is an expression$\varphi (n P) = \sum\limits_{r = 0}^d {n^r \varphi _r } (P, n)$, where the coefficientϕ(P, n) depends only on the congruence class ofn modulo the smallest positive integerk such that the affine hull of eachr-face ofk P is spanned by points ofΛ. Moreover,ϕ _r satisfies the Euler-type relation$\sum\limits_F {( - 1)^{\dim F} } \varphi _r (F, n) = ( - 1)^r \varphi _r ( - P, - n)$ where the sum extends over all non-empty facesF ofP. The proof involves a specific representation of simple such valuations, analogous to Hadwiger's representation of weakly continuous valuations on alld-polytopes. An example of particular interest is the lattice-point enumeratorG, whereG(P) = card(P∩λ); the results of this paper confirm conjectures of Ehrhart concerningG.