The Poincaré 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 Poincaré problem for foli...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2016-08
Main Authors: Genzmer, Yohann, Mol, Rogério
Format: Article
Language:English
Subjects:
Invariants
Poincare problem
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 \(\mathbb{P}^{2}_{\mathbb{C}}\), 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 \(\mathcal{F}\). 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, \({\rm deg}(S) \leq {\rm deg}(\mathcal{F}) + 2\).
ISSN:2331-8422