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 first author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non- representable hyperbolic matroid. The Va ́mos matroid and a generalization of it are to this day 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, due to Jordan, between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non-representable. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.
Highlights
Hyperbolic polynomials have their origin in PDE theory, they have during recent years been studied in diverse areas such as control theory, optimization, real algebraic geometry, probability theory, computer science and combinatorics, see [22, 23, 24, 25] and the references therein
A problem that has received considerable interest is the generalized Lax conjecture which asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices
To identify potential counterexamples to the generalized Lax conjecture, it is of interest to study hyperbolic matroids which are not representable over C, or even better not representable over any field
Summary
Hyperbolic polynomials have their origin in PDE theory, they have during recent years been studied in diverse areas such as control theory, optimization, real algebraic geometry, probability theory, computer science and combinatorics, see [22, 23, 24, 25] and the references therein. Choe et al [7] and Gurvits [13] proved that hyperbolic polynomials give rise to a class of matroids. This class of matroids, called hyperbolic matroids or matroids with the weak half-plane property, properly contains the class of matroids which are representable over the complex numbers, see [7]. To identify potential counterexamples to the generalized Lax conjecture, it is of interest to study hyperbolic matroids which are not representable over C, or even better not representable over any (skew) field. In this paper we first show that the Non-Pappus and Non-Desargues matroids are hyperbolic (but not representable over any field) by considering a well known connection between hyperbolic polynomials and Euclidean Jordan algebras. We refer to [1] for full proofs
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