Log-concavity for independent sets of valuated matroids
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916315791360000 |
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| author | Giansiracusa, Jeffrey Rincón, Felipe Schleis, Victoria Ulirsch, Martin |
| author_facet | Giansiracusa, Jeffrey Rincón, Felipe Schleis, Victoria Ulirsch, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05808 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Log-concavity for independent sets of valuated matroids Giansiracusa, Jeffrey Rincón, Felipe Schleis, Victoria Ulirsch, Martin Combinatorics Algebraic Geometry 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. |
| title | Log-concavity for independent sets of valuated matroids |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/2407.05808 |