Lattice invariant valuations on rational polytopes

Let be a lattice ind-dimensional euclidean space , and the rational vector space it generates. If is a valuation invariant under, andP is a polytope with vertices in , then for non-negative integersn there is an expression , where the coefficient(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 of. Moreover, _r satisfies the Euler-type relation 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); the results of this paper confirm conjectures of Ehrhart concerningG.