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

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

2007-06-07

arxiv.org/abs/0706.1092v1

Mathematics

Combinatorics

21 pages

Scientific paper

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.

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