Journal of Dynamical and Control Systems

, Volume 5, Issue 1, pp 1–50 | Cite as

Integration of Complex Differential Equations

  • B. Azevedo Scárdua


Let X be a polynomial vector field in ℂ2; then it defines an algebraic foliation \(\mathcal{F}\) on ℂP(2). If \(\mathcal{F}\) admits a Liouvillian first integral on ℂP(2), then it is transversely affine outside some algebraic invariant curve S⊂ ℂP(2). If, moreover, for some irreducible component S0 ⊂ S, the singularities q ∈ Sing \(\mathcal{F}\) ∪ S are generic, then either \(\mathcal{F}\) is given by a closed rational 1-form or it is a rational pull-back from a Bernoulli foliation \({\mathcal{R}}:p(x)dy - (y^2 a(x) + yb(x))dx = 0{ on }\bar {\mathbb{C}} \times \bar {\mathbb{C}}.\) This result has several applications such as the study of foliations with algebraic limit sets on ℂP(2)(2), the classification polynomial complete vector fields over ℂ2, and topological rigidity of foliations on ℂP(2). We also address the problem of moderate integration for germs of complex ordinary differential equations.

Holomorphic foliation holonomy Liouvillian function affine transverse structure 


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Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • B. Azevedo Scárdua
    • 1
  1. 1.Inst. Mat. Pura AplicadaRio de JaneiroBrazil

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