Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues

Combot, Thierry

doi:10.5802/aif.3063

Integrable planar homogeneous potentials of degree

- 1

with small eigenvalues
[Les potentiels intégrables homogènes de degré

- 1

du plan avec petites valeurs propres]

Combot, Thierry ¹

¹ IMB, Universié de Bourgogne 9 avenue Alain Savary 21078 Dijon Cedex (France)

Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2253-2298.

Résumé
Abstract

On démontre une classification complète des potentiels $V$ méromorphiquement intégrables homogènes de degré $- 1$ , analytiques rééls sur $ℝ^{2} ∖ {0}$ . Dans le cas plus général où $V$ est seulement méromorphe sur un ouvert d’une variété algébrique, on démontre une classification de tous les potentiels intégrables ayant un point de Darboux $c$ tel que $V^{'} (c) = - c, c_{1}^{2} + c_{2}^{2} \neq 0$ et $Sp (\nabla^{2} V (c)) \subset {- 1, 0, 2}$ . Enfin, on présente une conjecture pour les autres valeurs propres et le cas des points de Darboux dégénérés $V^{'} (c) = 0$ .

We give a complete classification of meromorphically integrable homogeneous potentials $V$ of degree $- 1$ which are real analytic on $ℝ^{2} ∖ {0}$ . In the more general case when $V$ is only meromorphic on an open set of an algebraic variety, we give a classification of all integrable potentials having a Darboux point $c$ with $V^{'} (c) = - c, c_{1}^{2} + c_{2}^{2} \neq 0$ and $Sp (\nabla^{2} V (c)) \subset {- 1, 0, 2}$ . We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case $V^{'} (c) = 0$ .

Reçu le : 2013-02-19
Révisé le : 2014-11-25
Accepté le : 2015-01-22
Publié le : 2016-10-04

DOI : 10.5802/aif.3063

Classification : 37J30
Keywords: Morales-Ramis theory, homogeneous potentials, D-finiteness, higher variational equations
Mot clés : Théorie de Morales-Ramis, potentiels homogènes, D-finitude, équations variationelles supérieures

Affiliations des auteurs :

Combot, Thierry ¹

¹ IMB, Universié de Bourgogne 9 avenue Alain Savary 21078 Dijon Cedex (France)

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     title = {Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues},
     journal = {Annales de l'Institut Fourier},
     pages = {2253--2298},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Combot, Thierry. Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues. Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2253-2298. doi : 10.5802/aif.3063. https://aif.centre-mersenne.org/articles/10.5802/aif.3063/

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