Ergodic Theory and Dynamical Systems

Table of Contents - 2001 - Volume 21, Issue 03  


Holomorphic foliations with Liouvillian first integrals

C. CAMACHO a1 and B. AZEVEDO SCÁRDUA a2
a1 Instituto de Matemática Pura e Aplicada, Estrada D. Castorina, 110 Jardim Botânico, Rio de Janeiro – RJ, CEP: 22460-320, Brazil
a2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro – RJ, Brazil

Abstract

Intuitively, a Liouvillian function on \mathbb{C} P(n) is one which is obtained from rational functions by a finite process of integrations, exponentiations and algebraic operations. This paper is devoted to the study of foliations determined by polynomial 1-forms which have a Liouvillian first integral. Our main result states that, under some mild restrictions on the singularities of the foliation, such a foliation must be either a linear foliation or an exponent two Bernoulli foliation after some rational pull-back. This proves that the highest level of transcendence for the ordinary differential equations which can be integrated by the use of elementary functions is reached at the Riccati equations.

(Received May 11 1999)
(Accepted June 26 2000)