Integration of Complex Differential Equations

Abstract

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.