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October, 2018 Local polar invariants and the Poincaré problem in the dicritical case
Yohann GENZMER, Rogério MOL
J. Math. Soc. Japan 70(4): 1419-1451 (October, 2018). DOI: 10.2969/jmsj/76227622


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 to 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, $\deg(S) \leq \deg (\mathcal{F}) + 2$.

Funding Statement

This work was supported by MATH-AmSud Project CNRS/CAPES/Concytec. The first author was supported by a grant ANR-13-JS01-0002-0. The second author was supported by Pronex/FAPERJ and Universal/CNPq.


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Yohann GENZMER. Rogério MOL. "Local polar invariants and the Poincaré problem in the dicritical case." J. Math. Soc. Japan 70 (4) 1419 - 1451, October, 2018.


Received: 11 October 2016; Revised: 1 April 2017; Published: October, 2018
First available in Project Euclid: 1 October 2018

MathSciNet: MR3868212
zbMATH: 07009707
Digital Object Identifier: 10.2969/jmsj/76227622

Primary: 32S65

Keywords: $GSV$-index , holomorphic foliation , invariant curves , Poincaré problem

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 4 • October, 2018
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