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Approximate roots, toric resolutions and deformations of a plane branch. (English) Zbl 1258.14039

Summary: We analyze the expansions in terms of the approximate roots of a Weierstrass polynomial \(f \in C \{x\}[y]\), 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 \(f\), 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.

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
32S10 Invariants of analytic local rings
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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