The Galois groupoid of \(P_{1}\) and its irreducibility
The author addresses the irreducibility of the Galois groupoid of Painlevé P1 equation \[ y'' = 6 y^2 + x. \] The Galois groupoid is defined according to the theory of Malgrange, and the first section of the paper is devoted to a swift and efficient presentation of that theory. Then, the machinery is applied to the foliation defined by P1 on \(\mathbb{C}^3\), and a notion of irreducibility is defined, motivated and proved. The relation with old attacks on the subject (in particular by Poincaré and Drach) is explained, as well as with other approaches to differential Galois theory (Kolchin, Umemura). Last, the author includes the proof of a theorem of Elie Cartan used in the text.
scientific article; zbMATH DE number 5311211