Math Colloquium: Hyperbolic Polynomials, Barrier functions & Determinantal Representations

Wednesday, June 8, 2022

3:00 PM-4:00 PM

Speaker: Victor Vinnikov (Ben-Gurion University of the Negev)

Abstract: A homogeneous polynomials with real coefficients is called hyperbolic with respect to a point if the restriction of the polynomial to any real line through the point is a single variable real polynomial with only real roots. Hyperbolic polynomials were first introduced by Garding in the 1950s (following the work of Petrovsky) in the context of linear hyperbolic PDEs with constant coefficients. In the last two decades hyperbolic polynomials and related hyperbolicity cones came to play a role in optimization providing good barrier functions for interior point methods. An important question here is whether the resulting hyperbolic programming is more general than semidefinite programming, equivalently whether the hyperbolicity of a polynomial can be always certified by an appropriate positive definite linear determinantal representation. In this talk I will give an overview of hyperbolic polynomials and their determinantal representations. I will also discuss their relation to complex polynomials that are stable with respect to a tubular domain and for which there also exist determinantal representations certifying stability.

Contact Information

Dr. Hugo Woerdeman
hjw27@drexel.edu