コロキウム

In this talk, based on the preprint https://arxiv.org/abs/1707.04446, we discuss the integrability of polynomial vector fields on the plane by means of the differential Galois theory. More concretely, using the variational equation around a particular solution, we obtain a necessary condition for the existence of a rational first integral. Our method is systematic: It is started with the first order variational equation and extended to the higher order one. The theoretical result is illustrated with several families of examples. A key point is to check whether 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 called 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 a connection of the non-integrablity with some higher transcendent functions, like the error function.