HYPERBOLIC POLYNOMIALS, INTERLACERS, AND SUMS OF SQUARES
Mario Kummer, Daniel Plaumann, Cynthia Vinzant · 2016
Abstract. Hyperbolic polynomials are real polynomials whose real hypersurfaces are max-imally nested ovaloids, the innermost 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 poly-nomials 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 hy-perbolicity 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 Vámos 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. 1.