Tree metrics and log-concavity for matroids

Abstract

We show that a set function $\hat{I}½$ satisfies the gross substitutes property if and only if its homogeneous generating polynomial $Z_{q,\hat{I}½}$ is a Lorentzian polynomial for all positive $q\le 1$, answering a question of Eur-Huh. We achieve this by giving a rank 1 upper bound for the distance matrix of an ultrametric tree, refining a classical result of Graham-Pollak. This characterization enables us to resolve two open problems that strengthen Mason's log-concavity conjectures for the number of independent sets of a matroid: one posed by Giansiracusa-Rincón-Schleis-Ulirsch for valuated matroids, and two posed by Dowling in 1980 and Zhao in 1985 for ordinary matroids.

17 pages; revised to add references; comments welcome