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.  A multi-variate version of the same polynomial is also known.

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 (multi-variate) Kolchin polynomial measuring the growth rate of d.

We can then consider the situation of an o-minimal structure K with a generic tuple d of commuting compatible derivations, and use the corresponding Kolchin polynomial to give a bound to the thorn rank of (K,d).

Joint work with E. Kaplan.