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Hyperbolic Polynomials and Convex Analysis
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- Journal:
- Canadian Journal of Mathematics / Volume 53 / Issue 3 / 01 June 2001
- Published online by Cambridge University Press:
- 20 November 2018, pp. 470-488
- Print publication:
- 01 June 2001
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- Article
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A homogeneous real polynomial
$p$ is hyperbolic with respect to a given vector 
$d$ if the univariate polynomial 
$t\,\mapsto \,p(x\,-\,td)$ 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.
