Hyperbolic Polynomials and Convex Analysis

Heinz H. Bauschke, Osman Güler, Adrian S. Lewis, Hristo S. Sendov
2001 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
A homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t → p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gårding 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 Gårding's result to arbitrary symmetric functions of the roots. Many classical
more » ... ts. 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.
doi:10.4153/cjm-2001-020-6 fatcat:iosi5qkhevas3lqd32uvwnzdpi