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Nuij Type Pencils of Hyperbolic Polynomials

Published online by Cambridge University Press:  20 November 2018

Krzysztof Kurdyka
Affiliation:
Laboratoire de Mathematiques (LAMA), Université Savoie Mont Blanc, UMR 5127 CNRS, 73-376 Le Bourget-du-Lac cedex France. e-mail: Krzysztof.Kurdyka@univ-savoie.fr
Laurentiu Paunescu
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. e-mail: laurent@maths.usyd.edu.au
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Abstract

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Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots), then $p+s{{p}^{'}}$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form ${{p}_{a}}(z,s)\,\,:=\,\,\,p\,(z)+\,\sum\nolimits_{k=1}^{d}{{{a}_{k}}{{s}^{k}}{{p}^{(k)}}(z)}$. We give a full characterization of those $a=({{a}_{1}},...,{{a}_{d}})\,\in \,{{\mathbb{R}}^{d}}$ for which ${{p}_{a}}(z,s)$ is a pencil of hyperbolic polynomials. We also give a full characterization of those $a=({{a}_{1}},...,{{a}_{d}})\,\in \,{{\mathbb{R}}^{d}}$ for which the associated families $ $ admit universal determinantal representations. In fact, we show that all these sequences come fromspecial symmetric Toeplitz matrices.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2017

References

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