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Holomorphic foliations with Liouvillian first integrals

Published online by Cambridge University Press:  04 June 2001

C. CAMACHO
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada D. Castorina, 110 Jardim Botânico, Rio de Janeiro – RJ, CEP: 22460-320, Brazil
B. AZEVEDO SCÁRDUA
Affiliation:
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.

Type
Research Article
Copyright
2001 Cambridge University Press

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