Holomorphic Foliations and Vector Fields with Degenerated Singularity in \((\mathbb {C}^2,0)\)

This work is focused on the local understanding of the solutions of differential equations in \((\mathbb {C}^{2},0)\) . These are Riemann surfaces whose behaviour is partially codified in a first stage with the first nonzero monomial of the series defining the vector field. From Poincaré to our days, the development of the theory from different perspectives has been so vast and the world of authors who have participated and enriched the theory is so wide, that we were forced to concentrate in this work only on those aspects related to the local analytical classification of vector fields with degenerate singularity (and their corresponding foliations), satisfying some genericity assumptions. In this way, we present a paper that addresses the problem of formal-analytic rigidity of vector fields and foliations with dicritical and non-dicritical singularities, their corresponding formal and analytic normal forms, and the analytic classification invariants—Thom’s invariants. We give some ideas concerning the proofs of some of the results presented here since we considered them relevant. In the last section we briefly mention a geometric perspective related to the analytic invariants, the exposition of which is contained in two recent works. It is our wish that the analysis of vector fields with larger degeneracies can be enriched by what is presented here.