Generalizations of Khovanskiĭ's theorem on growth of sumsets in abelian semigroups: (extended abstract)

We show that if P is a lattice polytope in the nonnegative orthant of Rk and χ is a coloring of the lattice points in the orthant such that the color χ(a+b) depends only on the colors χ(a) and χ(b), then the number of colors used on the lattice points lying in nP is for large n given by a polynomial (or, for rational P, by a quasipolynomial). This unifies a classical result of Ehrhart on lattice points in polytopes and a result of Khovanskiĭ on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanskiĭ's result.