Complete polynomial vector fields in two complex variables

Dominique Cerveau 1.  and Bruno Scárdua 2.
  • 1. Université de Rennes I—IRMAR, Campus de Beaulieu—35042— Rennes Cedex, France.
  • 2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP. 21945-970 Rio de Janeiro—RJ, Brasil.

Abstract

We study the classification of polynomial vector fields in two complex variables under the hypotheses that the singularities are isolated and the flow is complete. Normal forms are obtained for the case the generic orbit is diffeomorphic to ℂ. For the case the generic orbit is diffeomorphic to ℂ ∖ {0} and there is an affine singularity we classify the linear part of the vector field and prove the existence of entire linearization or first integral.

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