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Hyperbolic polynomials, interlacers and sums of squares

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Polynomial Optimisation

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 Vmos matroid shows that this relaxation is not always exact.

This talk is part of the Isaac Newton Institute Seminar Series series.

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