Hilbert polynomials for finitary matroids

Abstract

We consider a tuple $Φ= (ϕ_1,\ldots,ϕ_m)$ of commuting maps on a finitary matroid $X$. We show that if $Φ$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{ϕ_1^{r_1}\cdotsϕ_m^{r_m}(a):a \in A\text{ and }r_1+\cdots+r_m = t\}$ is eventually a polynomial in $t$ (we also give a multivariate version of the polynomial). This allows us easily recover Khovanskii's theorem on the growth of sumsets, the existence of the classical Hilbert polynomial, and the existence of the Kolchin polynomial. We also prove some new Kolchin polynomial results for differential exponential fields and derivations on o-minimal fields, as well as a new result on the growth of Betti numbers in simplicial complexes.