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

Jean-Pierre Kahane

Vladimir Petrov Kostov ¹

¹Laboratoire 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 (Original language)
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 $P * Q$ .

Article information
Cite this article

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

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

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     doi = {10.1016/j.crma.2007.10.003},
     language = {en},
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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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