Petter Brändén: A survey of the generalized Lax conjecture

A cone in Euclidean space which is linearly isomorphic to a linear section of the cone of positive semidefinite matrices is called spectrahedral. Spectrahedral cones are the sets of feasible solutions to semidefinite programming.

Hyperbolicity cones are basic semialgebraic convex cones associated to hyperbolic polynomials, and were first studied by Gårding. All spectrahedral cones are hyperbolicity cones, and the generalized Lax conjecture (GLC) states that the converse is true. There would be several benefits of a positive solution to GLC. It would provide an algebraic way of testing if a cone is spectrahedral. Conversely it would provide a better understanding of hyperbolicity cones.

In its original form, the Lax conjecture was stated by Peter Lax in 1958 for cones in R^3, and proved by Helton and Vinnikov in 2007.

For higher dimensional cones, the generalized Lax conjecture is still wide open. We will survey on different approaches to the Lax conjecture using methods from combinatorics and real algebraic geometry.