Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which gives the linear dimension of the graded pieces of a finitely generated graded algebra over a field. A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field grows eventually polynomially. 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. We present a common generalization of these three results in terms of finitary matroids, a structure coming from algebraic combinatorics and model theory. This is joint work with Antongiulio Fornasiero.