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arXiv · 2601.02547

Tree metrics and log-concavity for matroids

Federico Ardila-Mantilla·Sergio Cristancho·Graham Denham·Christopher Eur·June Huh·Botong Wang
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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.

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Federico Ardila-Mantilla, Sergio Cristancho, Graham Denham, Christopher Eur, June Huh, Botong Wang. 2026-01-05. Tree metrics and log-concavity for matroids. https://arxiv.org/abs/2601.02547

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