Séminaire de Géométrie, Groupes et Dynamiques

Juan Morales (Universidad Politécnica de Madrid), "Differential Galois Theory and non-Integrability of Planar Polynomial Vector Fields"

We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational equations around a particular solution, it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check wether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the Risch algorithm. In this way we point out the connection of the non integrablity with some higher transcendent functions, like the error function. This is a joint work with P.B. Acosta-Humánez, J.T. Lázaro and C. Pantazi.
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