The Lax Conjecture Is True

In 1958 Lax conjectured that hyperbolic polynomials in three variables 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 R of degree d (the maximum of the degrees of the monomials in the expansion of p). We call p homogeneous if p(tw) = tp(w) for all real t and vectors w ∈ R: equivalently, every monomial in the expansion of p has degree d. We denote the set of such polynomials by H(d). By identifying a polynomial with its vector of coefficients, we can consider H(d) as a normed vector space of dimension ( n+d−1 d ) . A polynomial p ∈ H(d) is hyperbolic with respect to a vector e ∈ R if p(e) = 0 and, for all vectors w ∈ R, the univariate polynomial t → p(w − te) has all real roots. The corresponding hyperbolicity cone is the open convex cone (see [5]) {w ∈ R : p(w − te) = 0 ⇒ t > 0}. For example, the polynomial w1w2 · · ·wn is hyperbolic with respect to the vector (1, 1, . . . , 1), since the polynomial t → (w1 − t)(w2 − t) · · · (wn − t) has roots w1, w2, . . . , wn; hence the corresponding hyperbolicity cone is the open positive orthant. Hyperbolic polynomials and their hyperbolicity cones originally appeared in the partial differential equations literature [4]. They have attracted attention more recently as fundamental objects in modern convex optimization [6, 1]. Three primary reasons drive this interest: (i) the definition of “hyperbolic polynomial” is strikingly simple; (ii) the class of hyperbolic polynomials, although not well-understood, is known to be rich — specifically, its interior in H(d) is nonempty; (iii) optimization problems posed over hyperbolicity cones, with linear objective and constraint functions, are amenable to efficient interior point algorithms. For more details on these reasons, see [6, 1]. In light of the interest of hyperbolic polynomials to optimization theorists, it is therefore natural to ask: how general is the class of hyperbolicity cones? In particular, do hyperbolicity cones provide a more general model for convex optimization Received by the editors April 2, 2003. 2000 Mathematics Subject Classification. Primary 15A45, 90C25, 52A41.