Testing hyperbolicity of real polynomials

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating hyperbolicity into nonnegativity conditions, which can then be tested via sum-of-squares relaxations.
Comment: 12 pages, 1 figure; minor changes in Section 2 and a few corrections