On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

Primitivo B. Acosta-Humánez; J. Tomás Lázaro; Juan J. Morales-Ruiz; Chara Pantazi

doi:10.3934/dcds.2015.35.1767

Article Contents

2015, Volume 35, Issue 5: 1767-1800. Doi: 10.3934/dcds.2015.35.1767

This issue Previous Article Next Article The magnetic ray transform on Anosov surfaces

On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

1.
Department of Mathematics, Universidad del Atlántico and Intelectual.Co, Barranquilla
2.
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona
3.
Department of Applied Mathematics, Technical University of Madrid, Madrid

Received: September 2013

Revised: September 2014

Published: May 2017

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Abstract

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.

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Mathematics Subject Classification: Primary: 12H05; Secondary: 32S65.

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