Symbolic Computations of First Integrals for Polynomial Vector Fields
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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 O(N ^ω+1), where N is the bound on the degree of a representation of the first integral and ω ∈ [2; 3] is the exponent of linear algebra. This result improves previous algorithms.
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