On the computation of Darboux first integrals of a class of planar polynomial vector fields

TL;DR

This paper develops an algorithm to compute Darboux first integrals for a specific class of planar polynomial vector fields, enhancing understanding of their integrability properties.

Contribution

It introduces an extended reduction procedure and an algorithm for computing Darboux first integrals in polynomial vector fields with particular geometric features.

Findings

Algorithm successfully computes Darboux first integrals for generic exponents.

Extended reduction procedure simplifies the analysis of these vector fields.

Provides a systematic method for integrability analysis in this class.

Abstract

We study the class of planar polynomial vector fields admitting Darboux first integrals of the type $\prod_{i=1}^r f_i^{\alpha_i}$, where the $\alpha_i$'s are positive real numbers and the $f_i$'s are polynomials defining curves with only one place at infinity. We show that these vector fields have an extended reduction procedure and give an algorithm which, from a part of the extended reduction of the vector field, computes a Darboux first integral for generic exponents.