Logic Seminar

Elliot Kaplan, McMaster University

"Hilbert polynomials for finitary matroids"

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). Time permitting, I’ll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) and how these polynomials can be used to bound model-theoretic ranks (like thorn-rank). This is joint work with Antongiulio Fornasiero.

MC 5479