Hyperbolic Polynomials and Convex Analysis
- 1 June 2001
- journal article
- Published by Canadian Mathematical Society in Canadian Journal of Mathematics
- Vol. 53 (3), 470-488
- https://doi.org/10.4153/cjm-2001-020-6
Abstract
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 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.Keywords
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