The Schur–Szegö composition for real polynomials

Vladimir Petrov Kostov

doi:10.1016/j.crma.2008.01.015

Mathematical Analysis

The Schur–Szegö composition for real 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 346 (2008) no. 5-6, pp. 271-276.

Abstract
Résumé

For two real polynomials in one variable $P = \sum_{j = 0}^{n} p_{j} x^{j}$ , $Q = \sum_{j = 0}^{n} q_{j} x^{j}$ set $W : = \sum_{j = 0}^{n} (p_{j} q_{j} / C_{n + k}^{j}) x^{j}$ where $k \in N \cup 0$ . For $k = 0$ this is the composition of Schur-Szegö of P and Q. We discuss the question if the numbers of negative, positive and complex roots of P and Q are known, what these numbers can be for W.

On définit d'après deux polynômes réels à une variable $P = \sum_{j = 0}^{n} p_{j} x^{j}$ , $Q = \sum_{j = 0}^{n} q_{j} x^{j}$ le polynôme $W : = \sum_{j = 0}^{n} (p_{j} q_{j} / C_{n + k}^{j}) x^{j}$ où $k \in N \cup 0$ . Pour $k = 0$ on obtient la composition de Schur-Szegö de P et Q Nous discutons la question si le nombre de racines strictement négatives, strictement positives et complexes de P et Q sont connus, quels peuvent être ces nombres pour W.

Received: 2008-01-03
Accepted: 2008-01-29
Published online: 2008-02-15

DOI: 10.1016/j.crma.2008.01.015

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 real polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {271--276},
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     volume = {346},
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     year = {2008},
     doi = {10.1016/j.crma.2008.01.015},
     language = {en},
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Vladimir Petrov Kostov. The Schur–Szegö composition for real polynomials. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 271-276. doi : 10.1016/j.crma.2008.01.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.01.015/

References
Cited by

[1] T. Craven; G. Csordas Composition theorems, multiplier sequences and complex zero decreasing sequences, Value distribution theory and related topics, Adv. Complex Anal. Appl., vol. 3, Kluwer Acad. Publ., Boston, MA, 2004, pp. 131-166

[2] V.P. Kostov The Schur–Szegö composition for hyperbolic polynomials, C. R. Acad. Sci. Paris, Ser. I, Volume 345 (2007) no. 9, pp. 483-488 | DOI

[3] 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

Cited by Sources:

^⁎ Research partially supported by research project 20682 for cooperation between CNRS and FAPESP “Zeros of algebraic polynomials”.

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