Clifford Realizations of Hyperbolic Cubics

Abstract

We prove the generalized Lax conjecture for cubics in five variables: their hyperbolicity cones are spectrahedral. Our approach constructs spectrahedral realizations from completely positive maps on Clifford algebras. The construction passes through quaternionic determinantal representations of certain extremal cubics. Convexity of the set of Clifford-realizable cubics then yields the result for all cubics in five variables. The approach also covers all previously known cubic cases of the generalized Lax conjecture: forms in at most four variables and symmetric forms in arbitrarily many variables. We further establish a sufficient criterion for Clifford realizability in arbitrary dimension and use it to obtain a neighborhood of the origin in the cubic norm ball consisting of Clifford-realizable cubics. Finally, we prove spectrahedrality for sparse hyperbolic quartics in four variables.