Comptes Rendus
no. 11
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 Pereira1
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.

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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.10.002
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/

[1] A.R. Amir-Moéz; A. Horn Singular values of a matrix, Amer. Math. Monthly, Volume 65 (1958), pp. 742-748

[2] G. Bilodeau Generating polynomials all of whose roots are real, Math. Mag., Volume 64 (1991), pp. 263-270

[3] J. Borcea (Convexity of twisted root maps, Rocky Mountain Math. J., in press. Available at) | arXiv

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

[5] K. Fan On a theorem of Weyl concerning eigenvalues of linear transformations I, Proc. Nat. Acad. Sci. USA, Volume 35 (1949), pp. 652-655

[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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