Séminaire Lotharingien de Combinatoire, 95B.53 (2026), 12 pp.
Federico Ardila-Mantilla, Sergio Cristancho, Graham Denham, Christopher Eur, June Huh and Botong Wang
Tree Metrics and Log-Concavity for Matroids
Abstract.
We characterize M♮-concave functions on sets in
terms of Lorentzian polynomials.
We achieve this by showing that a certain rank 1 modification
of the distance matrix of a ultrametric tree is positive
semi-definite, generalizing classical results of Schoenberg and of
Graham and Pollak on tree metrics.
Using the characterization, we resolve two open questions that extend
Mason’s log-concavity conjectures for the number of independent sets
of a matroid: one posed by Giansiracusa, Rincón, Schleis, and
Ulirsch for valuated matroids, and another by Dowling and Zhao for
ordinary matroids.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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