Very hyperbolic and stably hyperbolic polynomials

Vladimir Petrov Kostov

doi:10.1016/j.crma.2004.05.010

Mathematical Analysis/Algebraic Geometry

Very hyperbolic and stably hyperbolic polynomials

Vladimir Arnol'd

Vladimir Petrov Kostov ¹

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

Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 157-162

Abstract (Original language)
Résumé

A real polynomial in one variable is hyperbolic if it has only real roots. A hyperbolic polynomial is very hyperbolic if it has hyperbolic primitives of all orders. A polynomial P is stably hyperbolic if $x^{k} P + Q$ is hyperbolic for suitable $k \in N$ and Q (polynomial of degree $⩽ k - 1$ ). We present some geometric properties of the domains of very hyperbolic and of stably hyperbolic polynomials in the family $x^{n} + a_{1} x^{n - 1} + \dots + a_{n}$ .

Un polynôme réel d'une variable est hyperbolique si toutes ses racines sont réelles. Un polynôme hyperbolique est très hyperbolique s'il a des primitives hyperboliques de tout ordre. Un polynôme P est stablement hyperbolique si $x^{k} P + Q$ est hyperbolique pour certains $k \in N$ et Q (polynôme de degré $⩽ k - 1$ ). Nous présentons des propriétés géométriques des domaines de polynômes très hyperboliques et stablement hyperboliques dans la famille $x^{n} + a_{1} x^{n - 1} + \dots + a_{n}$ .

Article information
Cite this article

Received: 2004-05-14
Accepted: 2004-05-18
Published online: 2004-07-23

DOI: 10.1016/j.crma.2004.05.010

Author's affiliations:

Vladimir Petrov Kostov ¹

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

Vladimir Petrov Kostov. Very hyperbolic and stably hyperbolic polynomials. Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 157-162. doi: 10.1016/j.crma.2004.05.010

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References
Cited by

[1] V.P. Kostov On the hyperbolicity domain of the polynomial $x^{n} + a_{1} x^{n - 1} + \dots + a_{n}$ , Serdica Math. J., Volume 25 (1999) no. 1, pp. 47-70

[2] V.P. Kostov, Very hyperbolic polynomials in one variable, Manuscript, 10 p

[3] V.P. Kostov, Very hyperbolic polynomials, Funct. Anal. Appl., in press

[4] V.P. Kostov On the geometric properties of Vandermonde's mapping and on the problem of moments, Proc. Roy. Soc. Edinburgh, Volume 112 (1989) no. 3–4, pp. 203-211

[5] I. Meguerditchian, Géométrie du discriminant réel et des polynômes hyperboliques, Thèse de doctorat, Univ. de Rennes I, soutenue le 24.01.1991

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