Convexity properties of twisted root maps

Abstract

The strong spectral order induces a natural partial ordering on the manifold $H_n$ of monic hyperbolic polynomials of degree $n$. We prove that twisted root maps associated with linear operators acting on $H_n$ are Gårding convex on every polynomial pencil and we characterize the class of polynomial pencils of logarithmic derivative type by means of the strong spectral order. Let $A^{\prime }$ be the monoid of linear operators that preserve hyperbolicity as well as root sums. We show that any polynomial in $H_n$ is the global minimum of its $A^{\prime }$-orbit and we conjecture a similar result for complex polynomials.

final version, to appear in Rocky Mountain J. Math.; 14 pages, no figures, LaTeX2e

References in corpus (1)

• Spectral order and isotonic differential operators of Laguerre-Polya type

Cited by in corpus (1)

• Choquet order for spectra of higher Lame operators and orthogonal polynomials