Local polar invariants and the Poincaré problem in the dicritical case

Abstract

We develop a study on local polar invariants of planar complex analytic foliations at (ℂ²,0), which leads to the characterization of second type foliations and of generalized curve foliations, as well as to a description of the 𝐺𝑆𝑉-index. We apply it to the Poincaré problem for foliations on the complex projective plane ℙ²_ℂ, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve 𝑆 in terms of the degree of the foliation ℱ. We characterize the existence of a solution for the Poincaré problem in terms of the structure of the set of local separatrices of ℱ over the curve 𝑆. Our method, in particular, recovers the known solution for the non-dicritical case, deg(𝑆) ≤ deg(ℱ) + 2.