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Nuij Type Pencils of Hyperbolic Polynomials

  Published:2017-01-13
 Printed: Sep 2017
  • Krzysztof Kurdyka,
    Laboratoire de Mathematiques (LAMA), Université Savoie Mont Blanc, UMR 5127 CNRS, 73-376 Le Bourget-du-Lac cedex France
  • Laurentiu Paunescu,
    School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
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Abstract

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e. has only real roots) then $p+sp'$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$. We give a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic polynomials. We give also a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$ admit universal determinantal representations. In fact we show that all these sequences come from special symmetric Toeplitz matrices.
Keywords: hyperbolic polynomial, stable polynomial, determinantal representa- tion, symmetric Toeplitz matrix hyperbolic polynomial, stable polynomial, determinantal representa- tion, symmetric Toeplitz matrix
MSC Classifications: 15A15, 30C10, 47A56 show english descriptions Determinants, permanents, other special matrix functions [See also 19B10, 19B14]
Polynomials
Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
15A15 - Determinants, permanents, other special matrix functions [See also 19B10, 19B14]
30C10 - Polynomials
47A56 - Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
 

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