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 or Riccati 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 [Formula omitted], where N is the bound on the degree of a representation of the first integral and [Formula omitted] 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.
Author(s): Guillaume Chèze [sup.1], Thierry Combot [sup.2] Author Affiliations: (1) grid.11417.32, 0000 0001 2353 1689, Institut de Mathématiques de Toulouse, UMR 5219, CNRS, UPS IMT, Université de Toulouse, , 118 [...]