Hyperbolic polynomials and convex analysis

A homogeneous real polynomial p is hyperbolic with respect to a given vector ri if the univariate polynomial t --> p(x - td) has all real roots for all vectors x. Motivated by partial differential equations, Garding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize Garding's result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.