Hilbert polynomials for finitary matroids

Let d be a finite tuple of commuting derivations on a field K. A classical result allows us to associate a numerical polynomial to d (the Kolchin polynomial), measuring the "growth rate" of d. We show that we can abstract from the setting of fields with derivations, and consider instead a matroid with a tuple d of commuting (quasi)-endomorphisms. In this setting too there exists a polynomial measuring the growth rate of d. Joint work with E. Kaplan