Hilbert polynomials for finitary matroids

TL;DR

The paper unifies Hilbert, Kolchin, and Khovanskii-type polynomial-growth phenomena using finitary matroids and commuting operator tuples. It introduces triangular and quasi-triangular systems, along with the $\Phi$-closure and $\Phi_*$-closure, and proves that the rank of operator-orbits of finite sets is eventually governed by polynomials (dimension and cumulative dimension polynomials), with multivariate generalizations tied to partitions. The authors derive numerous classical recoveries and new results across tropical geometry, simplicial Betti numbers, and difference-differential/o-minimal model theory, illustrating the framework’s broad applicability. The approach hinges on generating-function rationality for decreasing rank functions and yields a versatile toolkit for analyzing growth in combinatorics, algebraic geometry, and model theory, linking closure operations to polynomial invariants in a unified way.

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.

Theorems & Definitions (68)

• Theorem
• Theorem A
• Corollary A
• Theorem B
• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Proposition 1.3
• proof