Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects

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 Br\"and\'en 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.