Computational Variants of the General Lax Conjecture

Abstract

Hyperbolicity cones are convex semialgebraic sets generalizing both polyhedral and spectrahedral cones, the latter forming the basic geometric sets from linear and semidefinite programming. Hyperbolic polynomials, which give rise to these hyperbolicity cones, have recently found applications in several areas of mathematics, statistical physics, computer science, and optimization. The general Lax conjecture is a fundamental question in real algebraic geometry and optimization: do hyperbolicity cones form a strict generalization of spectrahedral cones?

In this talk, we will give an introduction to hyperbolic polynomials and their cones, and raise several computational questions related to these objects, which blend algebraic complexity, real algebraic geometry, proof complexity and optimization.