Non-oscillating solutions of a differential equation and Hardy fields

François Blais[1]; Robert Moussu[1]; Fernando Sanz[2]

  • [1] Université de Bourgogne IMB UFR Sciences et Tecniques 9 Avenue Alain Savary, BP47870 21004 Dijon cedex (France)
  • [2] Universidad de Valladolid Depto. de Álgebra, Geometría y Topología Facultad de Ciencias E-47005 Valladolid (Spain)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 6, page 1825-1838
  • ISSN: 0373-0956

Abstract

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Let ϕ : x ↦ ϕ ( x ) , x ≫ 0 be a solution of an algebraic differential equation of order n , P ( x , y , y ′ , ... , y ( n ) ) = 0 . We establish a geometric criterion so that the germs at infinity of ϕ and the identity function on ℝ belong to a common Hardy field. This criterion is based on the concept of non oscillation.

How to cite

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Blais, François, Moussu, Robert, and Sanz, Fernando. "Solutions non oscillantes d’une équation différentielle et corps de Hardy." Annales de l’institut Fourier 57.6 (2007): 1825-1838. <http://eudml.org/doc/10278>.

@article{Blais2007,
abstract = {Soit $\varphi :x\mapsto \varphi (x), x\gg 0$ une solution à l’infini d’une équation différentielle algébrique d’ordre $n$, $P(x,y,y^\{\prime\},\ldots ,y^\{(n)\})=0$. Nous donnons un critère géométrique pour que les germes à l’infini de $\varphi $ et de la fonction identité sur $\{\mathbb\{R\}\}$ appartiennent à un même corps de Hardy. Ce critère repose sur le concept de non oscillation.},
affiliation = {Université de Bourgogne IMB UFR Sciences et Tecniques 9 Avenue Alain Savary, BP47870 21004 Dijon cedex (France); Université de Bourgogne IMB UFR Sciences et Tecniques 9 Avenue Alain Savary, BP47870 21004 Dijon cedex (France); Universidad de Valladolid Depto. de Álgebra, Geometría y Topología Facultad de Ciencias E-47005 Valladolid (Spain)},
author = {Blais, François, Moussu, Robert, Sanz, Fernando},
journal = {Annales de l’institut Fourier},
keywords = {oscillation; Hardy field; semi-algebraic; pfaffian},
language = {fre},
number = {6},
pages = {1825-1838},
publisher = {Association des Annales de l’institut Fourier},
title = {Solutions non oscillantes d’une équation différentielle et corps de Hardy},
url = {http://eudml.org/doc/10278},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Blais, François
AU - Moussu, Robert
AU - Sanz, Fernando
TI - Solutions non oscillantes d’une équation différentielle et corps de Hardy
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 6
SP - 1825
EP - 1838
AB - Soit $\varphi :x\mapsto \varphi (x), x\gg 0$ une solution à l’infini d’une équation différentielle algébrique d’ordre $n$, $P(x,y,y^{\prime},\ldots ,y^{(n)})=0$. Nous donnons un critère géométrique pour que les germes à l’infini de $\varphi $ et de la fonction identité sur ${\mathbb{R}}$ appartiennent à un même corps de Hardy. Ce critère repose sur le concept de non oscillation.
LA - fre
KW - oscillation; Hardy field; semi-algebraic; pfaffian
UR - http://eudml.org/doc/10278
ER -

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