Hyperbolic polynomials and the Kadison-Singer problem

Abstract

Recently Marcus, Spielman and Srivastava gave a spectacular proof of a theorem which implies a positive solution to the Kadison-Singer problem via Weaver's $K{S}_r$ conjecture. We extend this theorem to the realm of hyperbolic polynomials and hyperbolicity cones, as well as to arbitrary ranks. We also sharpen the theorem by providing better bounds, which imply better bounds in Weaver's $K{S}_r$ conjecture for each $r>2$. For $r=2$ our bound agrees with Bownik et al.

Parts of this work are based on unpublished lecture notes arXiv:1412.0245

References in corpus (1)

• Hyperbolic polynomials and the Marcus-Spielman-Srivastava theorem

Cited by in corpus (2)

• Improved bounds in Weaver's ${\mathrm{KS}}_r$ conjecture for high rank positive semidefinite matrices
• Upper and Lower bounds for matrix discrepancy