We give a complete classification of meromorphically integrable homogeneous potentials of degree which are real analytic on . In the more general case when is only meromorphic on an open set of an algebraic variety, we give a classification of all integrable potentials having a Darboux point with and . We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case .
On démontre une classification complète des potentiels méromorphiquement intégrables homogènes de degré , analytiques rééls sur . Dans le cas plus général où est seulement méromorphe sur un ouvert d’une variété algébrique, on démontre une classification de tous les potentiels intégrables ayant un point de Darboux tel que et . Enfin, on présente une conjecture pour les autres valeurs propres et le cas des points de Darboux dégénérés .
Revised:
Accepted:
Published online:
Classification: 37J30
Keywords: Morales-Ramis theory, homogeneous potentials, D-finiteness, higher variational equations
@article{AIF_2016__66_6_2253_0, author = {Combot, Thierry}, 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}, volume = {66}, number = {6}, year = {2016}, doi = {10.5802/aif.3063}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3063/} }
TY - JOUR TI - Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues JO - Annales de l'Institut Fourier PY - 2016 DA - 2016/// SP - 2253 EP - 2298 VL - 66 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3063/ UR - https://doi.org/10.5802/aif.3063 DO - 10.5802/aif.3063 LA - en ID - AIF_2016__66_6_2253_0 ER -
%0 Journal Article %T Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues %J Annales de l'Institut Fourier %D 2016 %P 2253-2298 %V 66 %N 6 %I Association des Annales de l’institut Fourier %U https://doi.org/10.5802/aif.3063 %R 10.5802/aif.3063 %G en %F AIF_2016__66_6_2253_0
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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