Symbolic Computations of First Integrals for Polynomial Vector Fields

Guillaume Chèze (1) , Thierry Combot (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IMB - Institut de Mathématiques de Bourgogne [Dijon]

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.

Domaines: Calcul formel [cs.SC]; Analyse classique [math.CA]; Systèmes dynamiques [math.DS]; Systèmes Solubles et Intégrables [nlin.SI]; Complexité [cs.CC]

Guillaume Chèze, Thierry Combot. Symbolic Computations of First Integrals for Polynomial Vector Fields. 2017. ⟨hal-01619911v1⟩