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

TL;DR

This paper investigates the integrability of polynomial vector fields in the plane using Picard-Vessiot Galois theory, focusing on Riccati foliations and specific families like quadratic, Liénard, and special function-related equations.

Contribution

It applies Galois theory to determine integrability conditions for polynomial fields, including new results for quadratic, Liénard, and special function equations.

Findings

Identifies integrability conditions for quadratic vector fields.

Provides new insights into Liénard equations' integrability.

Analyzes the Poincaré problem for certain families.

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, Li\'enard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincar\'e problem for some families is also approached.