Session B3 - Symbolic Analysis
July 15, 17:30 ~ 17:55 - Room B2
Differential Galois Theory and non-integrability of planar polynomial vector fields
Juan J. Morales-Ruiz
Universidad Politécnica de Madrid, Spain - firstname.lastname@example.org
We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, as a corollary of a previous result with Ramis and Simó on Hamiltonian systems, it is proved that a necessary condition for the existence of a meromorphic first integral is that the identity component of the Galois group of the higher order variational equations around a particular solution must be abelian. We illustrate this theorem with several families of examples. A key point in these applications is to check wether a suitable primitive is or not elementary. 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 will point out the connection of the non integrablity of polynomial fields with some higher transcendent functions, like the error function.
Joint work with Primitivo B. Acosta-Humánez (Universidad Simón Bolivar, Colombia), J. Tomás Lázaro (Universitat Politècnica de Catalunya, Spain) and Chara Pantazi (Universitat Politècnica de Catalunya, Spain).