Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects
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| Autores principales: | , , , |
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| Formato: | Preprint |
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2025
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| _version_ | 1866909834656350208 |
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| author | Baker, Matthew Huh, June Kummer, Mario Lorscheid, Oliver |
| author_facet | Baker, Matthew Huh, June Kummer, Mario Lorscheid, Oliver |
| contents | Lorentzian polynomials serve as a bridge between continuous and discrete convexity, connecting analysis and combinatorics. In this article, we study the topology of the space $\mathbb{P}\textrm{L}_J$ of Lorentzian polynomials on $J$ modulo $\mathbb{R}_{>0}$, which is nonempty if and only if $J$ is the set of bases of a polymatroid. We prove that $\mathbb{P}\textrm{L}_J$ is a manifold with boundary of dimension equal to the Tutte rank of $J$, and more precisely, that it is homeomorphic to a closed Euclidean ball with the Dressian of $J$ removed from its boundary. Furthermore, we show that $\mathbb{P}\textrm{L}_J$ is homeomorphic to the thin Schubert cell $\textrm{Gr}_J(\mathbb{T}_q)$ of $J$ over the triangular hyperfield $\mathbb{T}_q$, introduced by Viro in the context of tropical geometry and Maslov dequantization, for any $q>0$. This identification enables us to apply the representation theory of polymatroids developed in a companion paper, as well as earlier work by the first and fourth authors on foundations of matroids, to give a simple explicit description of $\mathbb{P}\textrm{L}_J$ up to homeomorphism in several key cases. Our results show that $\mathbb{P}\textrm{L}_J$ always admits a compactification homeomorphic to a closed Euclidean ball. They can also be used to answer a question of Brändén in the negative by showing that the closure of $\mathbb{P}\textrm{L}_J$ within the space of all polynomials modulo $\mathbb{R}_{>0}$ is not homeomorphic to a closed Euclidean ball in general. In addition, we introduce the Hausdorff compactification of the space of rescaling classes of Lorentzian polynomials and show that the Chow quotient of a complex Grassmannian maps naturally to this compactification. This provides a geometric framework that connects the asymptotic structure of the space of Lorentzian polynomials with classical constructions in algebraic geometry. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_02907 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects Baker, Matthew Huh, June Kummer, Mario Lorscheid, Oliver Combinatorics Algebraic Geometry Lorentzian polynomials serve as a bridge between continuous and discrete convexity, connecting analysis and combinatorics. In this article, we study the topology of the space $\mathbb{P}\textrm{L}_J$ of Lorentzian polynomials on $J$ modulo $\mathbb{R}_{>0}$, which is nonempty if and only if $J$ is the set of bases of a polymatroid. We prove that $\mathbb{P}\textrm{L}_J$ is a manifold with boundary of dimension equal to the Tutte rank of $J$, and more precisely, that it is homeomorphic to a closed Euclidean ball with the Dressian of $J$ removed from its boundary. Furthermore, we show that $\mathbb{P}\textrm{L}_J$ is homeomorphic to the thin Schubert cell $\textrm{Gr}_J(\mathbb{T}_q)$ of $J$ over the triangular hyperfield $\mathbb{T}_q$, introduced by Viro in the context of tropical geometry and Maslov dequantization, for any $q>0$. This identification enables us to apply the representation theory of polymatroids developed in a companion paper, as well as earlier work by the first and fourth authors on foundations of matroids, to give a simple explicit description of $\mathbb{P}\textrm{L}_J$ up to homeomorphism in several key cases. Our results show that $\mathbb{P}\textrm{L}_J$ always admits a compactification homeomorphic to a closed Euclidean ball. They can also be used to answer a question of Brändén in the negative by showing that the closure of $\mathbb{P}\textrm{L}_J$ within the space of all polynomials modulo $\mathbb{R}_{>0}$ is not homeomorphic to a closed Euclidean ball in general. In addition, we introduce the Hausdorff compactification of the space of rescaling classes of Lorentzian polynomials and show that the Chow quotient of a complex Grassmannian maps naturally to this compactification. This provides a geometric framework that connects the asymptotic structure of the space of Lorentzian polynomials with classical constructions in algebraic geometry. |
| title | Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/2508.02907 |