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

摘要

We study the class of planar polynomial vector fields admitting Darboux first integrals of the type Pi(r)(i=1) 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. (C) 2019 Elsevier Inc. All rights reserved.