Lorentzian polynomials and matroids over triangular hyperfields 2: Analytic aspects

Abstract

Brändén and Huh showed that Lorentzian polynomials unify Hodge-Riemann relations in combinatorics: their supports are M-convex, and every M-convex set supports a Lorentzian polynomial. Baker, Huh, Kummer, and Lorscheid later proved that, for every $q>0$, the projectivized space $\mathbf{P}{L}_J$ of Lorentzian polynomials with support $J$ is homeomorphic to the thin Schubert cell ${\mathrm{Gr}}_J^{\mathrm{w}}({\mathbb{T}}_q)$ of weak representations of $J$ over the generalized triangular hyperfield ${\mathbb{T}}_q$. We study the quantitative relation between Lorentzian polynomials and representations over triangular hyperfields. For every matroid $M$, we prove that some $q>0$ depending on $M$ satisfies ${\mathrm{Gr}}_M^{\mathrm{w}}({\mathbb{T}}_q)\subseteq \mathbf{P}{L}_M\subseteq {\mathrm{Gr}}_M^{\mathrm{w}}({\mathbb{T}}_2)$. Thus $\mathbf{P}{L}_M$ lies between two thin Schubert cells, each homeomorphic to it. More generally, for every M-convex set $J$, some $q>0$ depending on $J$ satisfies $N{\mathrm{Gr}}_J^{\mathrm{w}}({\mathbb{T}}_q)\subseteq \mathbf{P}{L}_J\subseteq N{\mathrm{Gr}}_J^{\mathrm{w}}({\mathbb{T}}_2)$, where $N$ denotes normalization. We also study $q(M):=\sup \{q>0:{\mathrm{Gr}}_M^{\mathrm{w}}({\mathbb{T}}_q)\subseteq \mathbf{P}{L}_M\}$. For $q(n):=q(U_{2,n})$, we prove $q(4)=2$ and $q(5)={\log }_{2}3$, with matching upper and lower bounds of order $1/n$; hence $q(n)=\hat{I}˜(1/n)$, so in particular no universal positive lower bound for $q(n)$ exists.

88 pages