Simple Darboux points of polynomial planar vector fields

Abstract

We are interested in some aspects of the integrability of complex polynomial planar vector fields in finite form. Especially, in the case of simple Darboux points, we deduce the famous Baum–Bott formula from a kind of global residue theorem; our elementary proof essentially relies on Hilbert's Nullstellensatz. As a corollary of our result, we propose formulas relating the various integers involved in the Lagutinskii–Levelt procedure for a Darboux polynomial at the various Darboux points. In particular, from the whole set of our formulas, it is possible to deduce an upper bound on the degree of irreducible Darboux polynomials in classical cases; with respect to such applications, this corollary seems to provide an alternate tool to usual genus formulas. As many people do in these subjects, we illustrate our corollary by giving a new simple proof of the fact that the polynomial Jouanolou derivation y s ∂ x + z s ∂ y + x s ∂ z , with s⩾2, has no Darboux polynomial.