Log-concavity of matroid h-vectors and mixed Eulerian numbers

TL;DR

This paper proves a strengthened log-concavity property of matroid h-vectors by connecting Tutte polynomial computations with mixed intersection theory and Hodge-Riemann relations, advancing matroid theory.

Contribution

It introduces a novel approach linking Tutte polynomial calculations with mixed intersection numbers and Hodge-Riemann relations to strengthen log-concavity results for matroids.

Findings

Computed Tutte polynomial via mixed intersection numbers

Established a stronger log-concavity of matroid h-vectors

Improved upon Dawson's conjecture using Hodge-Riemann relations

Abstract

For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the $h$-vector of a matroid complex, improving on an old conjecture of Dawson.