Comptes Rendus
Mathematical Analysis/Algebraic Geometry
Very hyperbolic and stably hyperbolic polynomials
Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 157-162.

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 xkP+Q is hyperbolic for suitable kN and Q (polynomial of degree k1). We present some geometric properties of the domains of very hyperbolic and of stably hyperbolic polynomials in the family xn+a1xn1++an.

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 xkP+Q est hyperbolique pour certains kN et Q (polynôme de degré k1). Nous présentons des propriétés géométriques des domaines de polynômes très hyperboliques et stablement hyperboliques dans la famille xn+a1xn1++an.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.05.010
Vladimir Petrov Kostov 1

1 Laboratoire J.-A. Dieudonné, CNRS UMR 6621, université de Nice, parc Valrose, 06108 Nice cedex 2, France
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     title = {Very hyperbolic and stably hyperbolic polynomials},
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.010/

[1] V.P. Kostov On the hyperbolicity domain of the polynomial xn+a1xn1++an, 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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