Local polar invariants and the Poincaré problem in the dicritical case

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 com...

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Bibliographic Details
Published in:Journal of the Mathematical Society of Japan 2018-10, Vol.70 (4), p.1419-1451
Main Authors: GENZMER, Yohann, MOL, Rogério
Format: Article
Language:English
Subjects:
Complex Variables
Dynamical Systems
Mathematics
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Summary: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 P2C, 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.
ISSN:0025-5645
DOI:10.2969/jmsj/76227622