Hilbert polynomials for finitary matroids

Let be a tuple of commuting maps on a finitary matroid . If satisfies certain conditions, then, for any finite subset of , the rank of the partial orbit of under is eventually equal to a polynomial (there is also a multivariate version of the polynomial). This allows us to easily recover Khovanskii’s theorem on the growth of sumsets, the classical Hilbert polynomial, the Kolchin polynomial for differential fields, and the Hilbert polynomial for tropical ideals.
We can also prove some new Kolchin polynomial results for differential exponential fields and derivations on o-minimal fields, as well as
the polynomial growth of Betti numbers in a simplicial complex.
Preprint: https://arxiv.org/abs/2208.01560