Log-concavity for independent sets of valuated matroids

preprint Log-concavity for independent sets of valuated matroids Jeffrey Giansiracusa author Felipe Rincón author Victoria Schleis author Martin Ulirsch author 114697 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. 2024 eng arXiv:2407.05808 2407.05808 Giansiracusa, J., Rincón, F., Schleis, V., & Ulirsch, M. (2024). Log-concavity for independent sets of valuated matroids. In arXiv:2407.05808. J. Giansiracusa, F. Rincón, V. Schleis, and M. Ulirsch, “Log-concavity for independent sets of valuated matroids,” arXiv:2407.05808. 2024. Giansiracusa, Jeffrey, Felipe Rincón, Victoria Schleis, and Martin Ulirsch. “Log-Concavity for Independent Sets of Valuated Matroids.” ArXiv:2407.05808, 2024. J. Giansiracusa, F. Rincón, V. Schleis, M. Ulirsch, ArXiv:2407.05808 (2024). Giansiracusa, Jeffrey, et al. “Log-Concavity for Independent Sets of Valuated Matroids.” ArXiv:2407.05808, 2024. Giansiracusa J, Rincón F, Schleis V, Ulirsch M. Log-concavity for independent sets of valuated matroids. arXiv:240705808. Published online 2024. @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} } 663062026-07-08T06:22:23Z2026-07-08T06:22:38Z