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

Presented by: Vladimir Arnol'd

Vladimir Petrov Kostov ¹

¹ Laboratoire 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
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 $x^{n} + a_{1} x^{n - 1} + \dots + a_{n}$ .

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

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     author = {Vladimir Petrov Kostov},
     title = {Very hyperbolic and stably hyperbolic polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {157--162},
     publisher = {Elsevier},
     volume = {339},
     number = {3},
     year = {2004},
     doi = {10.1016/j.crma.2004.05.010},
     language = {en},
}

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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/

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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