Hilbert polynomials for finitary matroids

TL;DR

This paper studies the polynomial growth of ranks in finitary matroids under certain commuting maps, unifying and extending classical results like Khovanskii's theorem and Kolchin polynomials.

Contribution

It introduces conditions under which the rank function in finitary matroids is polynomial, generalizing several classical theorems and providing new results in differential algebra and topology.

Findings

Rank growth is polynomial under specified conditions

Unified proof of Khovanskii's theorem and Hilbert polynomial existence

New results on Betti number growth in simplicial complexes

Abstract

We consider a tuple $\Phi = (\phi_1,\ldots,\phi_m)$ of commuting maps on a finitary matroid $X$. We show that if $\Phi$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{\phi_1^{r_1}\cdots\phi_m^{r_m}(a):a \in A\text{ and }r_1+\cdots+r_m = t\}$ 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.