Tree metrics and log‐concavity for matroids
Federico Ardila‐Mantilla, Sergio Cristancho, Graham Denham, Christopher Eur, June Huh, Botong Wang
Source abstract
Abstract We show that a set function satisfies the gross substitutes property if and only if its homogeneous generating polynomial is a Lorentzian polynomial for all positive , 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 open problems that strengthen Mason's log‐concavity conjectures for the numbers 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.
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