On Polars of Plane Branches

TL;DR

This paper investigates the equisingularity classes of general polars of plane branches, providing a detailed description based on analytic classification, especially for branches with low multiplicity.

Contribution

It extends the understanding of how the equisingularity classes of polars depend on the analytic type of plane branches, offering explicit descriptions for classes with known classifications.

Findings

Describes the equisingularity classes of polars for branches with multiplicity ≤ 4.

Shows how to determine the classes using analytic classification methods.

Provides explicit examples for particular equisingularity classes.

Abstract

It is well known that the equisingularity class of the general polar of a plane branch is not the same for all branches in a given equisingularity class, but it is the same for sufficiently general ones and depends upon the analytic type of the branch. The aim of this paper is to go beyond generality and show how one could describe the equisingularity classes of (general) polars of all branches in a given equisingularity class, making use of the analytic classification of branches as described in previous work by the first two authors. We will show how this works in some particular equisingularity classes for which one has the complete explicit analytic classification, and in particular for all branches of multiplicity less or equal than four.