A note on the Darboux theory of integrability of non-autonomous polynomial differential systems

In this work, we unfold some differential algebraic aspects of Darboux first integrals of polynomial vector fields. An interesting improvement is that our approach can be applied both to autonomous and non-autonomous vector fields. We give a sufficient and necessary condition for the existence of a Darboux first integral of a specific form for a polynomial vector field with some known algebraic invariant hypersurfaces. For the autonomous case, the classical result of Darboux is obtained as a corollary. For the non-autonomous case our characterization improves a known criterium. (paper)