Hilbert polynomials for finitary matroids

Antongiulio Fornasiero and Elliot Kaplan

Vol. 333 (2024), No. 2, 273–308

DOI: 10.2140/pjm.2024.333.273

Abstract

We consider a tuple $\Phi =({\varphi }_1,\dots <!--FUNCTION\; APPLICATION-->,{\varphi }_m)$ 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 $\{{\varphi }_1^{r_1}\cdots {\varphi }_m^{r_m}(a):a\in A\text{ and }{r}_1+\cdots +r_m=t\}$ is eventually a polynomial in $t$ (we also give a multivariate version of the polynomial). This allows us to 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.

Keywords

finitary matroid, Hilbert polynomial, Kolchin polynomial, exponential field, o-minimal field, sumset

Mathematical Subject Classification

Primary: 05B35

Secondary: 03C64, 05E40, 12H05, 12H10, 13D40

Milestones

Received: 30 May 2024

Revised: 19 December 2024

Accepted: 22 December 2024

Published: 28 December 2024