Submission date: 2 August 2022

Abstract:

We consider a tuple Φ = (φ_1,...,\phi_m) of commuting maps on a finitary matroid X. We show that if Φ satisfies certain conditions, then for any finite set A ⊆ X, the rank of {φ_1^{r_1}...φ_m^{r_m}(a) : a in A and r_1+...+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.

Mathematics Subject Classification: Primary 05B35. Secondary 03C64, 05E40, 12H05, 12H10, 13D40

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Full text arXiv 2208.01560: pdf, ps.