Hyperbolic Polynomials and Generalized Clifford Algebras

TL;DR

This paper explores the realization of hyperbolicity cones as spectrahedra using generalized Clifford algebras, providing a new sufficient condition for spectrahedrality that can be verified via semidefinite programming.

Contribution

It introduces a novel approach connecting Clifford algebras to hyperbolic polynomials, offering a verifiable criterion for spectrahedral representation.

Findings

If -1 is not a sum of hermitian squares in the Clifford algebra, the hyperbolicity cone is spectrahedral.

The condition can be checked efficiently with a single semidefinite program.

The approach advances understanding of the generalized Lax conjecture.

Abstract

We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if -1 is not a sum of hermitian squares in the Clifford algebra of a hyperbolic polynomial, then its hyperbolicity cone is spectrahedral. Our result also has computational applications, since this sufficient condition can be checked with a single semidefinite program.