Tree metrics and log-concavity for matroids
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917202251218944 |
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| author | Ardila-Mantilla, Federico Cristancho, Sergio Denham, Graham Eur, Christopher Huh, June Wang, Botong |
| author_facet | Ardila-Mantilla, Federico Cristancho, Sergio Denham, Graham Eur, Christopher Huh, June Wang, Botong |
| contents | We show that a set function $ν$ satisfies the gross substitutes property if and only if its homogeneous generating polynomial $Z_{q,ν}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02547 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tree metrics and log-concavity for matroids Ardila-Mantilla, Federico Cristancho, Sergio Denham, Graham Eur, Christopher Huh, June Wang, Botong Combinatorics We show that a set function $ν$ satisfies the gross substitutes property if and only if its homogeneous generating polynomial $Z_{q,ν}$ 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. |
| title | Tree metrics and log-concavity for matroids |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.02547 |