Darboux theory of integrability in the sparse case

Darboux’s theorem and Jouanolou’s theorem deal with the existence of rst integrals and rational rst integrals of a polynomial vector eld. These results are given in terms of the degree of the polynomial vector eld. Here we show that we can get the same kind of results if we consider the size of a Newton polytope associated to the vector eld. Furthermore, we show that in this