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Hyperbolic polynomials, interlacers and sums of squares

Presented by: 
D Plaumann Universität Konstanz
Date: 
Friday 19th July 2013 - 09:30 to 10:00
Venue: 
INI Seminar Room 1
Abstract: 
A real polynomial is hyperbolic if it defines a hypersurface consisting of maximally nested ovaloids. These polynomials appear in many areas of mathematics, including convex optimisation, combinatorics and differential equations. We investigate the relation between a hyperbolic polynomial and the set of polynomials that interlace it. This set of interlacers is a convex cone, which we realize as a linear slice of the cone of nonnegative polynomials. We combine this with a sums-of-squares-relaxation to approximate a hyperbolicity cone explicitly by the projection of a spectrahdedron. A multiaffine example coming from the Vámos matroid shows that this relaxation is not always exact.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons