Generalizations of Khovanskii's theorems on growth of sumsets in abelian semigroups

TL;DR

This paper generalizes classical theorems on lattice points and sumsets, showing polynomial behavior in colorings of lattice points within dilated polytopes and strengthening multivariate sumset results.

Contribution

It unifies Ehrhart and Macdonald's lattice point results with Khovanskii's sumset theorems, providing new polynomial and quasipolynomial formulas and combinatorial proofs.

Findings

Number of colors in dilated polytopes is polynomial or quasipolynomial for large n.

Unified classical results on lattice points and sumsets under a common framework.

Provided combinatorial proofs for multivariate generalizations of Khovanskii's theorem.

Abstract

We show that if $P$ is a lattice polytope in the nonnegative orthant of $\R^k$ and $\chi$ is a coloring of the lattice points in the orthant such that the color $\chi(a+b)$ depends only on the colors $\chi(a)$ and $\chi(b)$, then the number of colors of the lattice points in the dilation $nP$ of $P$ is for large $n$ given by a polynomial (or, for rational $P$, by a quasipolynomial). This unifies a classical result of Ehrhart and Macdonald on lattice points in polytopes and a result of Khovanski\u\i{} on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanski\u\i's theorem. Another result of Khovanski\u\i{} states that the size of the image of a finite set after $n$ applications of mappings from a finite family of mutually commuting mappings is for large $n$ a polynomial. We give a combinatorial proof of a multivariate generalization of this theorem.