Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects

TL;DR

The paper develops a comprehensive topological framework for spaces of Lorentzian polynomials with fixed matroid support, establishing that ${\mathbb P}\mathrm{L}_J$ is a manifold with boundary and is homeomorphic to the thin Schubert cell ${\rm Gr}_J({\mathbb T}_q)$ for any $q>0$, with dimension given by the Tutte rank. It constructs and exploits a deep link between Lorentzian polynomials and matroid representations over triangular hyperfields, enabling explicit homeomorphism-type descriptions in key cases and enabling compactifications to closed Euclidean balls via Dressian rays. The authors introduce and relate several objects—Dressian, Dr_J, universal/trust foundations, and Maslov dequantization—to develop a robust geometry/ topology bridge, including Hausdorff and Chow-quotient perspectives that connect to tropical geometry and algebraic geometry. They provide explicit computations (e.g., Betsy Ross, $U_{2,4}$, and elliptic-type examples) showing when compactifications are ball-like and when they are not, and they derive Euler-characteristic formulas tying these topological features to combinatorial data from base polytopes and regular polymatroid subdivisions. Overall, the work yields a unifying topological picture for Lorentzian polynomials and matroid representations over tracts, with concrete consequences for compactifications, boundary structures, and connections to classical 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 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.

Figures (4)

• Figure 1: The picture shows the reduced Dressian $\mathop{\mathrm{\underline{Dr}}}\nolimits_M$ (red) inside the space $\mathop{\mathrm{\underline{L}}}\nolimits_M$ (purple) of orbits of Lorentzian polynomials in logarithmic coordinates when $M$ is the uniform matroid $U_{2,4}$. The boundary (green) is the image of $\mathop{\mathrm{\underline{Gr}}}\nolimits_M({\mathbb R})$ under the map from \ref{['part1_topology:eq:mapfromgrassmannian']}.
• Figure 2: Point-line arrangement of the Betsy Ross matroid
• Figure 3: A $2$-dimensional projection of the Petersen graph
• Figure 4: The deletion of ${\mathcal{T}}_{11}$ by $11$ is a configuration of ten points and ten lines in the plane such that each point is contained in three lines and each line contains three points. Its Dressian was studied in Brandt-Speyer22.

Theorems & Definitions (254)

• Theorem 1.1
• Example 1
• Theorem 1.2: \ref{['part1_topology:cor:lorismanifoldwithboundary']}
• Example 2
• Theorem 1.3: \ref{['part1_topology:thm:Alor']}
• Lemma 1: \ref{['part1_topology:lemma: petterlem']}
• Theorem 1.4: \ref{['part1_topology:thm:Alor']}
• Theorem 1.5
• Theorem 1.6: \ref{['part1_topology:cor:allarehomeo']}
• Conjecture 1