Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which eventually coincides with the linear dimension of the graded pieces of a finitely generated module over a polynomial ring.

A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field is eventually polynomial.

More recently, Khovanskii showed that for finite subsets \(A\) and \(B\) of a commutative semigroup, the size of the sumset \(A+tB\) is eventually polynomial in \(t\).

I will present a common generalization of these three results in terms of finitary matroids (also called pregeometries). I'll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) as well as some applications to bounding model-theoretic ranks.

This is joint work with Antongiulio Fornasiero.