Seeking Darboux polynomials

Abstract

We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in Rn. We apply these techniques to some well-known systems when n=2,3,4. We also propose a general method for computing Darboux polynomials in the plane. As an application we prove that a family of potential systems, that includes the van der Pol one, has no Darboux polynomials, giving in particular a new simple proof that the van der Pol limit cycle is not algebraic.