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

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.