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

Combot, Thierry ¹

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

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

Abstract
Résumé

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 $V^{'} (c) = 0$ .

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 $V^{'} (c) = 0$ .

Received: 2013-02-19
Revised: 2014-11-25
Accepted: 2015-01-22
Published online: 2016-10-04

DOI: 10.5802/aif.3063

Classification: 37J30
Keywords: Morales-Ramis theory, homogeneous potentials, D-finiteness, higher variational equations

Author's affiliations:

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},
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Combot, Thierry. Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues. Annales de l'Institut Fourier, Volume 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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