Approximate roots, toric resolutions and deformations of a plane branch

摘要

We analyze the expansions in terms of the approximate roots of a Weierstrass polynomial f ∈ C{x}[y/], defining a plane branch (C, 0), in the light of the toric embedded resolution of the branch. This leads to the definition of a class of (non-equisingular) deformations of a plane branch (C, 0) supported on certain monomials in the approximate roots of /, which are essential in the study of Harnack smoothings of real plane branches by Risler and the author. Our results provide also a geometrical approach to Abhyankar's irreducibility criterion for power series in two variables and also a criterion to determine if a family of plane curves is equisingular to a plane branch.