Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects

TL;DR

This paper explores the topology of Lorentzian polynomial spaces related to matroids over triangular hyperfields, revealing their manifold structure, homeomorphisms to Euclidean balls, and connections to tropical geometry and algebraic geometry.

Contribution

It characterizes the topological structure of Lorentzian polynomial spaces, linking them to tropical geometry and providing explicit descriptions and compactifications.

Findings

The space of Lorentzian polynomials is a manifold with boundary homeomorphic to a Euclidean ball.

Identifies the space with thin Schubert cells over triangular hyperfields.

Provides a compactification of the polynomial space and relates it to algebraic geometric constructions.

Abstract

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.