Toward Effective Liouvillian Integration

TL;DR

This paper investigates the properties of foliations on the projective plane with Liouvillian first integrals, establishing bounds on invariant algebraic curves and integrating factors based on the foliation's degree and singularities.

Contribution

It provides new bounds on invariant algebraic curves and integrating factors for foliations with Liouvillian integrals, extending understanding of their algebraic structure.

Findings

Invariant algebraic curves have degree bounded by a function of the foliation's degree.

Existence of a bound for the degree of the simplest integrating factor depending on the foliation.

Invariant algebraic curves of small degree exist for foliations with rational first integrals and intermediate Kodaira dimension.

Abstract

We prove that foliations on the projective plane admitting a Liouvillian first integral but not admitting a rational first integral always have invariant algebraic curves of degree bounded by a function of the degree of the foliation. We establish, for the same class of foliations, the existence of a bound for the degree of the simplest integrating factor depending only on the degree of the foliation and on the nature of its singularities. We also prove the existence of invariant algebraic curves of small degree for foliations with rational first integral and intermediate Kodaira dimension.