Le groupoïde de Galois de $P\_1$ et son irréductibilité

Abstract

In this article, the Galois groupoid of the first Painlevé equation is computed. This computation use E. Cartan's classification of structural equations of pseudogroups acting on $C^2$ and the degeneration of the first Painlevé equation on an elliptic equation ($y'' = 6y^2$). A definition of reducibility for singular holomorphic foliations is proposed. A characterisation of reducible foliations on their Galois groupoid is given and applied to prove the foliation irreducibility of the first Painlevé equation.