Hyperbolic and stable polynomials and their certifying determinantal representations

by Mr Victor Vinnikov (Ben-Gurion University)

Thursday Mar 13, 2025, 2:00 PM → 4:00 PM Europe/Paris

Hyperbolic polynomials were introduced by Garding (who was influenced by Petrovsky) in the 1950s in the study of linear PDES with constant coefficients. Hyperbolicity is essentially a homogeneous multivariable generalization of real upper half plane stability --- a real polynomial in one variable having no zeroes in the upper half plane and therefore having only real zeroes. A hyperbolic polynomial determines a convex cone called a hyperbolicity cone, and in the last two decades these came to attention in the context of interior point methods in optimization. In (up to) three variables, a hyperbolic polynomial always admits a (linear) determinantal representation certifying its hyperbolicity (Lax conjecture from 1950s that was established in the early 2000s). This implies that a hyperbolicity cone is a spectrahedral cone --- it is defined by a linear matrix inequality, and the corresponding hyperbolic programming is nothing else than semidefinite programming that was extensively developed and applied since before the turn of the century. The existence of certifying determinantal representations in higher dimension, called the generalized Lax conjecture, is so far wide open.

 

In this talk I will survey some of the known and the unknown about hyperbolic polynomials and hyperbolocity cones. I will then discuss some recent and ongoing joint work with Hugo Woerdeman (Drexel) relating hyperbolic polynomials to complex polynomials that are stable, i.e., have no zeroes, on a tube domain over a cone. In this case the existence of certifying determinantal representations can be (sometimes) established using ideas of multivariable operator theory such as realizations, finiteness of von Neumann norm, and hermitian Positivstellensaetze (reminiscent of the classical Positivstellensatetze of real algebraic geometry but using sums of hermitian  squares and evaluations on commuting operators).