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Darboux's theorem and Jouanolou's theorem deal with the existence of firstintegrals and rational first integrals of a polynomial vector field. Theseresults are given in terms of the degree of the polynomial vector field. Herewe show that we can get the same kind of results if we consider the size of aNewton polytope associated to the vector field. Furthermore, we show that inthis context the bound is optimal.

Darboux theory of integrability in the sparse case