Morales-Ramis Theorems via Malgrange pseudogroup

Casale, Guy

Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 2593-2610.

Dans cet article, nous montrons que les équations variationnelles le long d’une solution d’une équation différentielle intégrable par quadratures ont un groupe de Galois différentielle virtuellement résoluble. Dans le cas particulier des systèmes hamiltoniens intégrables au sens de Liouville la preuve redonne le théorème de Morales-Ramis-Simó. La preuve consiste à montrer que le groupe de Galois de l’équation variationnelle est un quotient d’un sous groupe d’un groupe d’isotropie du pseudogroupe de Malgrange de l’équation non linéaire. On relie ensuite les propriétés de ce groupe d’isotropie en un point spécial à celles du groupe d’isotropie au point générique en utilisant le théorème d’approximation d’Artin.

In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.

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