Toward Effective Liouvillian Integration
Algebraic Geometry
2018-04-20 v3 Classical Analysis and ODEs
Complex Variables
Dynamical Systems
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.
Keywords
Cite
@article{arxiv.1604.05276,
title = {Toward Effective Liouvillian Integration},
author = {Gaël Cousin and Alcides Lins Neto and Jorge Vitório Pereira},
journal= {arXiv preprint arXiv:1604.05276},
year = {2018}
}
Comments
Enhancement of the previous version, with one extra theorem (Theorem C), 35 pages