Seeking Darboux Polynomials

© 2014, Springer Science+Business Media Dordrecht. We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in $\mathbb {R}^{n}$. 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.