Newton-Puiseux algorithm and triple points for planes curves

TL;DR

This paper introduces an algorithmic approach to the Newton-Puiseux procedure for approximating branches of algebraic plane curves at singular points, illustrating its application on complex examples including triple points.

Contribution

It provides an accessible, algorithmic description of the Newton-Puiseux method and demonstrates its use in analyzing singularities like triple points on plane curves.

Findings

Effective polynomial approximations for curve branches at singularities.

Application to complex singularities such as points of multiplicity 6.

Insights into the structure of triple points on plane curves.

Abstract

The paper is an introduction to the use of the classical Newton-Puiseux procedure, oriented to an algorithmic description of it. This procedure enables to get polynomial approximations for parameterizations of branches of an algebraic plane curve at a singular point. We look for an approach that can be easily grasped and almost self contained. We illustrate the use of the algorithm, first in a completely worked out example of a curve with a point of multiplicity 6, and secondly in the study of triple points on reduced plane curves.