Integration of Complex Differential Equations

Abstract

Let X be a polynomial vector field in {\Bbb C}^2s then it defines an algebraic foliation {\cal F} on {\Bbb C}P(2). If {\cal F} admits a Liouvillian first integral on {\Bbb C}P(2), then it is transversely affine outside some algebraic invariant curve S\subset {\Bbb C}P(2). If, moreover, for some irreducible component S_0 \subset S, the singularities q ∈ Sing {\cal F} \cup S are generic, then either {\cal F} is given by a closed rational 1-form or it is a rational pull-back from a Bernoulli foliation {\cal R}: p(x)\thinspace dy-(y^2a(x)+yb(x))\thinspace dx=0 on \overline {\Bbb C} \times \overline {\Bbb C}. This result has several applications such as the study of foliations with algebraic limit sets on {\Bbb C}P(2)(2), the classification polynomial complete vector fields over {\Bbb C}^2, and topological rigidity of foliations on {\Bbb C}P(2). We also address the problem of moderate integration for germs of complex ordinary differential equations.