# On the factorization of the polar of a plane branch

## TL;DR

This paper provides a detailed factorization of the polar of a general plane branch, refining previous descriptions and characterizing classes with specific singularity behaviors, advancing understanding of complex plane curve singularities.

## Contribution

It offers the most complete factorization description of the polar of a general plane branch, extending prior results and characterizing equisingularity classes with unique polar component properties.

## Key findings

- Refined factorization of the polar of a plane branch.
- Characterization of classes with polar components having fewer characteristic exponents.
- Generalization of previous results for r=2 to broader classes.

## Abstract

In this paper we present the most complete description as possible of the factorization of the general polar of the general member of an equisingularity class of irreducible germs of complex plane curves. Our result will refine the rough description of the factorization given by M. Merle in the 70's and it is based on a result given by E. Casas-Alvero in the 90's that describes the cluster of the singularities of such polars. By using our analysis, it will be possible to characterize all equisingularity classes of irreducible plane germs with r characteristic exponents having the exceptional behavior that the general polar of a general curve in this equisingularity class has only irreducible components with less than r characteristic exponents, generalizing a result previously obtained for r=2 by the authors.