2 00 3 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. A homogeneous polynomial p on R n is hyperbolic with respect to a vector e ∈ R n if p(e) = 0 and, for all vectors w ∈ R n , the univariate polynomial t → p(w − te) has all real roots. The corresponding hyperbolicity cone is the open convex cone (see [5]) {w ∈ R n : p(w − te) = 0 ⇒ t > 0}. For example, the polynomial w 1 w 2 · · · w n is hyperbolic with respect to the vector (1, 1,. .. , 1), with hyperbolicity cone 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].