Non-representable hyperbolic matroids

TL;DR

This paper demonstrates that certain classical matroids, including Non-Pappus and Non-Desargues, are non-representable hyperbolic matroids, expanding the known examples and connecting hyperbolic polynomials with geometric and algebraic inequalities.

Contribution

It proves that specific classical matroids are non-representable hyperbolic matroids and identifies a broad class of such matroids, confirming a conjecture and strengthening inequalities related to hyperbolicity.

Findings

Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids.

A large class of hyperbolic matroids includes the Ve1mos and generalized Ve1mos matroids.

Strengthened inequalities such as Laguerre-Ture1n and Jensen's inequality.

Abstract

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the second author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non-representable hyperbolic matroid. The V\'amos matroid and a generalization of it are, prior to this work, the only known instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids which contains the V\'amos matroid and the generalized V\'amos matroids recently studied by Burton, Vinzant and Youm. This proves a conjecture of Burton et al. We also prove that many of the matroids considered here are non-representable. The proof of hyperbolicity for the matroids in the class depends on proving nonnegativity of certain symmetric polynomials. In particular we generalize and strengthen several inequalities in the literature, such as the Laguerre-Tur\'an inequality and Jensen's inequality. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.