  Hyperbolic polynomials are real polynomials whose real hypersurfaces are
nested ovaloids, the inner most of which is convex. These polynomials appear in
many areas of mathematics, including optimization, combinatorics and
differential equations. Here we investigate the special connection between a
hyperbolic polynomial and the set of polynomials that interlace it. This set of
interlacers is a convex cone, which we write as a linear slice of the cone of
nonnegative polynomials. In particular, this allows us to realize any
hyperbolicity cone as a slice of the cone of nonnegative polynomials. Using a
sums of squares relaxation, we then approximate a hyperbolicity cone by the
projection of a spectrahedron. A multiaffine example coming from the Vamos
matroid shows that this relaxation is not always exact. Using this theory, we
characterize the real stable multiaffine polynomials that have a definite
determinantal representation and construct one when it exists.
