Stable and Hyperbolic Polynomials and their Determinantal Representations (24rit023)

The Banff International Research Station will host the "Stable and Hyperbolic Polynomials and their Determinantal Representations" Research in Teams workshop in Banff from February 4 - 11, 2024.

Hyperbolic polynomials were originally introduced in the study of linear hyperbolic PDEs with constant coefficients in the 1950s. In the last several decades hyperbolic polynomials and associated hyperbolicity cones came to play a prominent role in several areas, in particular in optimization where they appear as feasibility sets for hyperbolic programming. An outstanding open problem, the generalized Lax conjecture, asks whether any hyperbolic polynomial admits, up to a factor, a positive definite linear determinantal representation certifying its hyperbolicity and representing the hyperbolicity cone as a spectrahedral cone (a linear slice of a cone of positive definite matrices) which is a feasibility set for semidefinite programming. We intend to progress towards the solution of this problem during the proposed research in teams at BIRS by relating hyperbolic polynomials to complex polynomials that are stable with respect to a tube domain over the hyperbolicity cone in the complex Euclidean space, and tackling certifying linear determinantal representations of complex stable polynomials using tools of multivariable operator theory. The progress in understanding such determinantal representations will be of considerable independentvalue to operator theory, real algebraic geometry, and optimization.