Spectrahedra and Ulrich Sheaves

The Generalized Lax Conjecture is a central open question in convex algebraic geometry. Its algebraic version states that for every hyperbolic polynomial h there is a polynomial q such that qh can be written as the determinant of a symmetric matrix with linear entries satisfying some positivity conditions. To that end we study Ulrich sheaves on reducible hypersurfaces. While we have not been able to prove the Generalized Lax Conjecture, we present some new results on ternary hyperbolic polynomials. This is joint work in progress with C. Hanselka as well as with S. Naldi and D. Plaumann.