Hyperbolic polynomials, interlacers, and sums of squares

Abstract

Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. Here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it. This set of interlacers is a convex cone, which we write as a linear slice of the cone of nonnegative polynomials. In particular, this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials. Using a sums of squares relaxation, we then approximate a hyperbolicity cone by the projection of a spectrahedron. A multiaffine example coming from the Vamos matroid shows that this relaxation is not always exact. Using this theory, we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists.

Minor corrections and improvements (20 pages, 11 figures)

References in corpus (1)

• Determinantal representations of hyperbolic plane curves: An elementary approach

Cited by in corpus (11)

• Geometry of free loci and factorization of noncommutative polynomials
• Determinantal Representations and Bézoutians
• A Note on the Hyperbolicity Cone of the Specialized Vámos Polynomial
• The Chow form of a reciprocal linear space
• Spectral linear matrix inequalities
• Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment
• Determinantal representations of semi-hyperbolic polynomials
• Hyperbolic Secant Varieties of M-Curves
• Spectrahedral representations of plane hyperbolic curves
• Matroids on Eight Elements with the Half-plane Property and Related Concepts
• Positive Ulrich Sheaves