Matrix-theoretical derivations of some results of Borcea–Shapiro on hyperbolic polynomials

Rajesh Pereira

doi:10.1016/j.crma.2005.10.002

Mathematical Analysis

Matrix-theoretical derivations of some results of Borcea–Shapiro on hyperbolic polynomials
[La dérivation matricielle de certaines résultats de Borcea–Shapiro sur les polynômes hyperboliques]

Présenté par : Jean-Pierre Kahane

Rajesh Pereira ¹

¹ Department of Mathematics and Statistics, University of Saskatchewan, S7N 5E6 Saskatoon SK, Canada

Comptes Rendus. Mathématique, Volume 341 (2005) no. 11, pp. 651-653.

Résumé
Abstract

On utilise l'analyse matricielle pour obtenir des démonstrations simples de deux résultats de Borcea–Shapiro sur la relation de majoration entre certains polynômes hyperboliques. On obtient aussi un résultat apparenté sur la majoration des zéros de polynômes complexes.

We use matrix analysis to give simple proofs of two theorems of Borcea–Shapiro which yield majorization relations between certain hyperbolic polynomials. We also prove a conjecture of Borcea involving majorization and the zeros of polynomials.

Reçu le : 2005-03-08
Accepté le : 2005-09-27
Publié le : 2005-11-07

DOI : 10.1016/j.crma.2005.10.002

Affiliations des auteurs :

Rajesh Pereira ¹

¹ Department of Mathematics and Statistics, University of Saskatchewan, S7N 5E6 Saskatoon SK, Canada

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Rajesh Pereira. Matrix-theoretical derivations of some results of Borcea–Shapiro on hyperbolic polynomials. Comptes Rendus. Mathématique, Volume 341 (2005) no. 11, pp. 651-653. doi : 10.1016/j.crma.2005.10.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.10.002/

Bibliographie
Cité par

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[4] J. Borcea; B. Shapiro Hyperbolic polynomials and spectral order, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 693-698

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[6] A.W. Marshall; I. Olkin Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979

[7] R. Pereira Differentiators and the geometry of polynomials, J. Math. Anal. Appl., Volume 285 (2003), pp. 336-348

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