, Volume 31, Issue 1, pp 509-516

Lattice invariant valuations on rational polytopes

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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(Pλ); the results of this paper confirm conjectures of Ehrhart concerningG.