Analytic classification of plane branches up to multiplicity 4

TL;DR

This paper classifies plane branches with multiplicity up to four analytically, providing a detailed understanding of their structure and equivalence classes.

Contribution

It offers the first complete analytic classification of plane branches with multiplicity four or less, filling a gap in singularity theory.

Findings

Complete classification for multiplicity 2, 3, and 4

Identification of invariants distinguishing classes

Framework for future classifications of higher multiplicities

Abstract

We perform the analytic classification of plane branches of multiplicity less or equal than four.