Symbolic Computations of First Integrals for Polynomial Vector Fields

In this article, we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian orRiccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in O (N omega+1), where N is the bound on the degree of a representation of the first integral and omega epsilon [2; 3] is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on the authors' websites. In the last section, we give some examples showing the efficiency of these algorithms.