no. 9
Mathematical Analysis
The Schur–Szegö composition for hyperbolic polynomials
[La composition de Schur–Szegö de polynômes hyperboliques]
Présenté par : Jean-Pierre Kahane
Kostov, Vladimir Petrov1
1 Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice, parc Valrose, 06108 Nice cedex 2, France
Comptes Rendus. Mathématique, Tome 345 (2007) no. 9, pp. 483-488

The composition of Schur–Szegö of the polynomials P(x)=∑j=0nCnjajxj and Q(x)=∑j=0nCnjbjxj is defined as P∗Q=∑j=0nCnjajbjxj. 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)=∑j=0nCnjajxj et Q(x)=∑j=0nCnjbjxj est définie comme P∗Q=∑j=0nCnjajbjxj. 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.10.003

Kostov, Vladimir Petrov 1

1 Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice, parc Valrose, 06108 Nice cedex 2, France 
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Kostov, Vladimir Petrov. The Schur–Szegö composition for hyperbolic polynomials. Comptes Rendus. Mathématique, Tome 345 (2007) no. 9, pp. 483-488. doi: 10.1016/j.crma.2007.10.003

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

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

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

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

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⁎ Research partially supported by Wenner-Grenn Foundation.