Hyperbolic polynomials and the Marcus-Spielman-Srivastava theorem

Abstract

Recently Marcus, Spielman and Srivastava gave a spectacular proof of a theorem which implies a positive solution to the Kadison-Singer problem. We extend (and slightly sharpen) this theorem to the realm of hyperbolic polynomials. A benefit of the extension is that the proof becomes coherent in its general form, and fits naturally in the theory of hyperbolic polynomials. We also study the sharpness of the bound in the theorem, and in the final section we describe how the hyperbolic Marcus-Spielman-Srivastava theorem may be interpreted in terms of strong Rayleigh measures. We use this to derive sufficient conditions for a weak half-plane property matroid to have k disjoint bases. This work is based on notes from a graduate course focused on hyperbolic polynomials and the recent papers of Marcus, Spielman and Srivastava, given by the author at the Royal Institute of Technology (Stockholm) in the fall of 2013.