Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz property

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From viewpoint combinatorial Hodge theory, it is natural to ask whether these are “strictly” log-concave simple In this paper, we show strictness graphic matroids, that is, Kirchhoff polynomials graphs strictly log-concave. Our key observation polynomial a complete graph can be seen as (irreducible) relative invariant certain prehomogeneous vector space, which may independently interesting in its own right. Furthermore, prove any ai∈R>0, a1x1+⋯+anxn∈RM1 satisfies strong Lefschetz property (moreover, Hodge–Riemann bilinear relation) at degree one Artinian Gorenstein algebra RM⁎ associated matroid M, defined by Maeno Numata