Abstract view
Nuij Type Pencils of Hyperbolic Polynomials


Published:20170113
Printed: Sep 2017
Krzysztof Kurdyka,
Laboratoire de Mathematiques (LAMA), Université Savoie Mont Blanc, UMR 5127 CNRS, 73376 Le BourgetduLac cedex France
Laurentiu Paunescu,
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
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.
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)
