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Convexity properties of twisted root maps

Julius Borcea
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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 $A'$ 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 $A'$-orbit and we conjecture a similar result for complex polynomials.

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Julius Borcea. 2007-10-20. Convexity properties of twisted root maps. https://arxiv.org/abs/math/0312321

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