The Poincar\'e problem in the dicritical case

We develop a study on local polar invariants of planar complex analytic foliations at $({\mathbb{C}}^2,0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as a description of the $GSV$-index. We apply it to the Poincar'e problem for foliations on the complex projective plane ${\mathbb{P}}_{\mathbb{C}}^2$, 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 $\mathcal{F}$. We characterize the existence of a solution for the Poincar'e problem in terms of the structure of the set of local separatrices of $\mathcal{F}$ over the curve $S$. Our method, in particular, recovers the known solution for the non-dicritical case, ${\rm deg}(S) \leq {\rm deg}($\mathcal{F}$) + 2$.