A generalization of weight polynomials to matroids

Trygve Johnsen, Jan Roksvold, Hugues Verdure

Published 2013-11-25, updated 2015-11-12Version 2

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid $M$. Our main result is that these polynomials are determined by Betti numbers associated with graded minimal free resolutions of the Stanley-Reisner ideals of $M$ and so-called elongations of $M$. Generalizing Greene's theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.