2026/07/16 by Matthew Baker, June Huh, Mario Kummer +1
#math.CO #math.MG
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 PLJ of Lorentzian polynomials with support J is homeomorphic to the thin Schubert cell GrwJ(\mathbbTq) of weak representations of J over the generalized triangular hyperfield \mathbbTq. 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 GrwM(\mathbbTq)\subseteqPLM\subseteqGrwM(\mathbbT2). Thus PLM 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 NGrwJ(\mathbbTq)\subseteqPLJ\subseteqNGrwJ(\mathbbT2), where N denotes normalization. We also study q(M):=sup\q>0:GrwM(\mathbbTq)\subseteqPLM\. For q(n):=q(U2,n), we prove q(4)=2 and q(5)=log2 3, with matching upper and lower bounds of order 1/n; hence q(n)=Θ(1/n), so in particular no universal positive lower bound for q(n) exists.