Multi-Harnack smoothings of real plane branches

TL;DR

The paper introduces a novel method for constructing smoothings of real plane branches using Viro Patchworking, focusing on multi-Harnack smoothings and their topological uniqueness related to complex equisingularity.

Contribution

It develops a new approach for smoothings of real plane branches via Newton non degenerate singularities and characterizes multi-Harnack smoothings by local data.

Findings

Unique topological type of multi-Harnack smoothings determined by complex equisingularity

Method extends Viro Patchworking to Newton degenerated singularities

Analysis of maximal position smoothings in the resolution process

Abstract

We introduce a new method for the construction of smoothings of a real plane branch $(C, 0)$ by using Viro Patchworking method. Since real plane branches are Newton degenerated in general, we cannot apply Viro Patchworking method directly. Instead we apply the Patchworking method for certain Newton non degenerate curve singularities with several branches. These singularities appear as a result of iterating deformations of the strict transforms of the branch at certain infinitely near points of the toric embedded resolution of singularities of $(C,0)$. We characterize the $M$-smoothings obtained by this method by the local data. In particular, we analyze the class of multi-Harnack smoothings, those smoothings arising in a sequence $M$-smoothings of the strict transforms of (C,0) which are in maximal position with respect to the coordinate lines. We prove that there is a unique the topological type of multi-Harnack smoothings, which is determined by the complex equisingularity type of the branch. This result is a local version of a recent Theorem of Mikhalkin.