  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.
