Hyperbolic polynomials, Strong Rayleigh matroids and the Marcus-Spielman-Srivastava theorem

: Hyperbolic polynomials are generalizations of determinantal polynomials, and hyperbolicity cones are generalizations of the cone of positive semidefinite matrices. 

I will show how the recent Marcus-Spielman-Srivastava theorem (implying the Kadison-Singer conjecture) may be generalized to hyperbolic polynomials, and point to some potential applications in combinatorics. 

The generalized Lax conjecture asserts that hyperbolicity cones are linear sections of the cone of positive semidefinite matrices. Recently the speaker disproved an algebraic strengthening of this conjecture by using Ingleton's inequality for matroids that are representable over some field. Kinser recently introduced an infinite family of inequalities that generalize Ingleton's inequality. For each Kinser inequality we construct a Strong Rayleigh matroid which fails to satisfy the inequality. This produces an infinite family of hyperbolic polynomials such that no power of a polynomial in the family is a determinantal polynomial. 

The second part of this talk is based on joint work with Nima Amini.  

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