Tree metrics and log-concavity for matroids

TL;DR

This paper characterizes when a set function satisfies the gross substitutes property using Lorentzian polynomials, and applies this to resolve open problems related to Mason's log-concavity conjectures for matroids.

Contribution

It provides a novel characterization linking Lorentzian polynomials to gross substitutes and solves longstanding open problems in matroid theory.

Findings

Set function $ u$ satisfies gross substitutes iff its polynomial $Z_{q, u}$ is Lorentzian for all $q \,\leq 1$.

Established a rank 1 upper bound for ultrametric tree distance matrices.

Resolved open problems on Mason's log-concavity conjectures for matroids.

Abstract

We show that a set function $\nu$ satisfies the gross substitutes property if and only if its homogeneous generating polynomial $Z_{q,\nu}$ 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-Rinc\'on-Schleis-Ulirsch for valuated matroids, and two posed by Dowling in 1980 and Zhao in 1985 for ordinary matroids.