---
_id: '66306'
abstract:
- lang: eng
  text: Recently, several proofs of the Mason--Welsh conjecture for matroids have
    been found, which asserts the log-concavity of the sequence that counts independent
    sets of a given size. In this article we use the theory of Lorentzian polynomials,
    developed by Brändén and Huh, to prove a generalization of the Mason-Welsh conjecture
    to the context of valuated matroids. In fact, we provide a log-concavity result
    in the more general setting of valuated discrete polymatroids, or equivalently,
    M-convex functions. Our approach is via the construction of a generic extension
    of a valuated matroid or M-convex function, so that the bases of the extension
    are related to the independent sets of the original matroid. We also provide a
    similar log-concavity result for valuated bimatroids, which, we believe, might
    be of independent interest.
author:
- first_name: Jeffrey
  full_name: Giansiracusa, Jeffrey
  last_name: Giansiracusa
- first_name: Felipe
  full_name: Rincón, Felipe
  last_name: Rincón
- first_name: Victoria
  full_name: Schleis, Victoria
  last_name: Schleis
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Giansiracusa J, Rincón F, Schleis V, Ulirsch M. Log-concavity for independent
    sets of valuated matroids. <i>arXiv:240705808</i>. Published online 2024.
  apa: Giansiracusa, J., Rincón, F., Schleis, V., &#38; Ulirsch, M. (2024). Log-concavity
    for independent sets of valuated matroids. In <i>arXiv:2407.05808</i>.
  bibtex: '@article{Giansiracusa_Rincón_Schleis_Ulirsch_2024, title={Log-concavity
    for independent sets of valuated matroids}, journal={arXiv:2407.05808}, author={Giansiracusa,
    Jeffrey and Rincón, Felipe and Schleis, Victoria and Ulirsch, Martin}, year={2024}
    }'
  chicago: Giansiracusa, Jeffrey, Felipe Rincón, Victoria Schleis, and Martin Ulirsch.
    “Log-Concavity for Independent Sets of Valuated Matroids.” <i>ArXiv:2407.05808</i>,
    2024.
  ieee: J. Giansiracusa, F. Rincón, V. Schleis, and M. Ulirsch, “Log-concavity for
    independent sets of valuated matroids,” <i>arXiv:2407.05808</i>. 2024.
  mla: Giansiracusa, Jeffrey, et al. “Log-Concavity for Independent Sets of Valuated
    Matroids.” <i>ArXiv:2407.05808</i>, 2024.
  short: J. Giansiracusa, F. Rincón, V. Schleis, M. Ulirsch, ArXiv:2407.05808 (2024).
date_created: 2026-07-08T06:22:23Z
date_updated: 2026-07-08T06:22:38Z
external_id:
  arxiv:
  - '2407.05808'
language:
- iso: eng
publication: arXiv:2407.05808
status: public
title: Log-concavity for independent sets of valuated matroids
type: preprint
user_id: '82981'
year: '2024'
...