The Schur–Szegö composition for hyperbolic polynomials

Vladimir Petrov Kostov

doi:10.1016/j.crma.2007.10.003

Mathematical Analysis

The Schur–Szegö composition for hyperbolic polynomials

Presented by: Jean-Pierre Kahane

Vladimir Petrov Kostov ¹

¹ Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice, parc Valrose, 06108 Nice cedex 2, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 483-488.

Abstract
Résumé

The composition of Schur–Szegö of the polynomials $P (x) = \sum_{j = 0}^{n} C_{n}^{j} a_{j} x^{j}$ and $Q (x) = \sum_{j = 0}^{n} C_{n}^{j} b_{j} x^{j}$ is defined as $P * Q = \sum_{j = 0}^{n} C_{n}^{j} a_{j} b_{j} x^{j}$ . In the case when P and Q are hyperbolic, i.e. with real roots only, we give the exhaustive answer to the question if the numbers of positive, negative and zero roots of P and Q are known what these numbers can be for $P * Q$ .

La composition de Schur–Szegö des polynômes $P (x) = \sum_{j = 0}^{n} C_{n}^{j} a_{j} x^{j}$ et $Q (x) = \sum_{j = 0}^{n} C_{n}^{j} b_{j} x^{j}$ est définie comme $P * Q = \sum_{j = 0}^{n} C_{n}^{j} a_{j} b_{j} x^{j}$ . Dans le cas où P et Q sont hyperboliques, c. à d. n'ayant que des racines réelles, nous donnons la réponse exhaustive à la question si on connaît les nombres de racines positives, négatives et nulles de P et Q, quels peuvent être ces nombres pour $P * Q$ .

Received: 2007-04-11
Accepted: 2007-09-28
Published online: 2007-10-29

DOI: 10.1016/j.crma.2007.10.003

Author's affiliations:

Vladimir Petrov Kostov ¹

¹ Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice, parc Valrose, 06108 Nice cedex 2, France

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     title = {The {Schur{\textendash}Szeg\"o} composition for hyperbolic polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {483--488},
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     number = {9},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.003},
     language = {en},
}

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Vladimir Petrov Kostov. The Schur–Szegö composition for hyperbolic polynomials. Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 483-488. doi : 10.1016/j.crma.2007.10.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.003/

References
Cited by

[1] S. Alkhatib, V.P. Kostov, The Schur–Szegö composition of real polynomials of degree 2, Revista Mat. Complut., in press

[2] V.P. Kostov; B.Z. Shapiro On the Schur–Szegö composition of polynomials, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 81-86

[3] V.P. Kostov; B.Z. Shapiro On arrangements of roots for a real hyperbolic polynomial and its derivatives, Bull. Sci. Math., Volume 126 (2002) no. 1, pp. 45-60

[4] V. Prasolov Polynomials, Algorithms and Computation in Mathematics, vol. 11, Springer-Verlag, Berlin, 2004 (xiv+301 pp)

Cited by Sources:

^⁎ Research partially supported by Wenner-Grenn Foundation.

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