Strictness of the log-concavity of generating polynomials of matroids

Abstract

Recently, it was proved by Anari-Oveis Gharan-Vinzant, Anari-Liu-Oveis Gharan-Vinzant and Brändén-Huh that, for any matroid M , its basis generating polynomial and its independent set generating polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities on matroids including a solution of strong Mason's conjecture. In this paper, we study the strictness of the log-concavity of these polynomials and determine when equality holds in these combinatorial inequalities. We also consider a generalization of our result to morphisms of matroids.

On Artinian Gorenstein algebras associated to the face posets of regular polyhedra

“…In [10], Nagaoka and Yazawa show that the algebra defined by a Kirchhoff polynomial, a homogeneous polynomial defined by a graph, satisfies the strong Lefschetz property and Hodge-Riemann relation "at degree one" (it will be explained in Remark 3.2) on the positive orthant. More generally, in [9], Murai, Nagaoka and Yazawa show that the algebra defined by the basis generating polynomial, a generalization of a Kirchhoff polynomial, satisfies the strong Lefschetz property and Hodge-Riemann relation at degree one on the positive orthant. Remark 3.2.…”

“…More precisely, F is log-concave on the positive orthant if and only if the Hessian matrix H F (a) of F has exactly one positive eigenvalue for a in the positive orthant. In [9], it was shown that the polynomials are strictly log-concave on the positive orthant, equivalently, the Hessian matrices have exactly one positive eigenvalue and are not degenerate. For an Artinian Gorenstein algebra associated to the polynomial, it was also shown that the strong Lefschetz property and Hodge-Riemann relation at degree one.…”

The eigenvalues of the Hessian matrices of the generating functions for trees with $k$ components

“…That is the Hessian matrix has exactly one positive eigenvalue and its Hessian does not vanish. We gives another proof of the theorem in the case of the truncated matroids of graphic matroids of the complete and complete bipartite graphs by directly calculation of the eigenvalues of the Hessian matrix in [8]. See also Remark 3.11. This paper is organized as follows: In Section 2, we consider the generating function for the forests.…”

“…The Hessian matrix of the generating function for forests with one components has exactly one positive eigenvalue and its Hessian does not vanish. More general, in [9] and [8], the Hessian matrix has exactly one positive eigenvalue and its Hessian does not vanish for the generating function for any simple graphic matroid and any simple matroid, respectively. Our main theorem is that for the generating function for forests with k components, its Hessian matrix and its Hessian are in the same situation.…”

“…Remark 3.11. It is shown that the Hessian of the generating function for simple matroid does not vanish in [8]. Moreover, its Hessian matrix has exactly one positive eigenvalue.…”