The Poincaré problem in the dicritical case

Abstract

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 Poincaré 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 Poincaré 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, $\mathrm{deg}(S)\le \mathrm{deg}(\mathcal{F})+2$.

31 pages, 2 figures

References in corpus (1)

• Modules formels locaux de feuilletages holomorphes

Cited by in corpus (1)

• Characterization of second type plane foliations using Newton polygons