Smooth Hyperbolicity Cones are Spectrahedral Shadows

TL;DR

This paper proves that all smooth hyperbolicity cones can be represented as spectrahedral shadows, advancing the understanding of their geometric structure and supporting the conjecture that all hyperbolicity cones are spectrahedral.

Contribution

It establishes that every smooth hyperbolicity cone is a spectrahedral shadow, providing a significant partial confirmation of the spectrahedrality conjecture.

Findings

Every smooth hyperbolicity cone is a spectrahedral shadow.

Supports the conjecture that all hyperbolicity cones are spectrahedral.

Advances geometric understanding of hyperbolicity cones.

Abstract

Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone, that is, a spectrahedral shadow.