A mapping connected with the Schur–Szegő composition

Vladimir Petrov Kostov

doi:10.1016/j.crma.2009.10.025

Mathematical Analysis

A mapping connected with the Schur–Szegő composition

Presented by: Jean-Pierre Kahane

Vladimir Petrov Kostov ¹

¹ Université de Nice, Laboratoire de Mathématiques, UMR 6621, parc Valrose, 06108 Nice cedex 2, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1355-1360

Abstract (Original language)
Résumé

Every monic polynomial in one variable of the form $(x + 1) S$ , $\deg S = n - 1$ , is presentable in a unique way as a Schur–Szegő composition of $n - 1$ polynomials of the form ${(x + 1)}^{n - 1} (x + a_{i})$ . We prove geometric properties of the affine mapping associating to the coefficients of S the $(n - 1)$ -tuple of values of the elementary symmetric functions of the numbers $a_{i}$ .

Tout polynôme unitaire à une variable de la forme $(x + 1) S$ , $\deg S = n - 1$ , est présentable de façon unique comme composition de Schur–Szegő de $n - 1$ polynômes ${(x + 1)}^{n - 1} (x + a_{i})$ . Nous prouvons des propriétés géométriques de l'application affine associant aux coefficients de S le $(n - 1)$ -uplet des valeurs des fonctions symétriques élémentaires des nombres $a_{i}$ .

Received: 2008-08-07
Accepted: 2009-10-27
Published online: 2009-11-07

DOI: 10.1016/j.crma.2009.10.025

Author's affiliations:

Vladimir Petrov Kostov ¹

¹ Université de Nice, Laboratoire de Mathématiques, UMR 6621, parc Valrose, 06108 Nice cedex 2, France

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     author = {Vladimir Petrov Kostov},
     title = {A mapping connected with the {Schur{\textendash}Szeg\H{o}} composition},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1355--1360},
     year = {2009},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2009.10.025},
     language = {en},
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Vladimir Petrov Kostov. A mapping connected with the Schur–Szegő composition. Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1355-1360. doi: 10.1016/j.crma.2009.10.025

References
Cited by

[1] S. Alkhatib; V.P. Kostov The Schur–Szegő composition of real polynomials of degree 2, Rev. Mat. Complut., Volume 21 (2008), pp. 191-206

[2] V.P. Kostov The Schur–Szegő composition for hyperbolic polynomials, C. R. Acad. Sci. Paris Sér. I, Volume 345 (2007), pp. 483-488

[3] V.P. Kostov Eigenvectors in the context of the Schur–Szegő composition of polynomials, Math. Balkanica, Volume 22 (2008) no. 1–2, pp. 155-173

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

[5] Victor Prasolov Polynomials, Algorithms and Computation in Mathematics, vol. 11, Springer-Verlag, Berlin, 2004

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