The Lax Conjecture Is True
Adrian S. Lewis, Pablo A. Parrilo, M. V. Ramana, Communicated Jonathan M. Borwein · 2003
Abstract. In 1958 Lax conjectured that hyperbolic polynomials in three vari-ables are determinants of linear combinations of three symmetric matrices. This conjecture is equivalent to a recent observation of Helton and Vinnikov. Consider a polynomial p on Rn of degree d (the maximum of the degrees of the monomials in the expansion of p). We call p homogeneous if p(tw) = tdp(w) for all real t and vectors w ∈ Rn: equivalently, every monomial in the expansion of p has degree d. We denote the set of such polynomials by Hn(d). By identifying a polynomial with its vector of coefficients, we can consider Hn(d) as a normed vector space of dimension n+d−1