Approximate Roots, Toric Resolutions and Deformations of a Plane Branch

TL;DR

This paper explores the relationship between approximate roots, toric resolutions, and deformations of plane branches, providing new formulas and criteria for understanding their singularities and equisingularity conditions.

Contribution

It introduces a novel class of non-equisingular deformations supported on approximate roots and derives a Kouchnirenko type formula for the Milnor number of plane branches.

Findings

Derived a Kouchnirenko type formula for Milnor number

Defined a new class of deformations supported on approximate roots

Provided an algorithmic equisingularity criterion

Abstract

We analyze the expansions in terms of the approximate roots of a Weierstrass polynomial $f$ 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 $f$. As a consequence we find out a Kouchnirenko type formula for the Milnor number $(C,0)$. Our results provide a geometrical approach to Abhyankar's straight line conditions and its consequences. As an application we give an equisingularity criterion for a family of plane curves to be equisingular to a plane branch and we express it algorithmically.