Nuij Type Pencils of Hyperbolic Polynomials

Kurdyka, Krzysztof ; Paunescu, Laurentiu

Canadian mathematical bulletin, Tome 60 (2017) no. 3, pp. 561-570

Résumé

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.

Détail

DOI : 10.4153/CMB-2016-079-1

Mots-clés : 15A15, 30C10, 47A56, hyperbolic polynomial, stable polynomial, determinantal representation, symmetric Toeplitz matrix