Singular Points of Real Quintic Curves Via Computer Algebra

TL;DR

This paper uses computer algebra to classify all types of real and complex singular points of quintic curves, providing a complete and verified taxonomy through symbolic computations.

Contribution

It offers the first complete classification of singular points for real and complex quintic curves using Maple, with a novel approach based on Puiseux expansions.

Findings

42 types of real singular points for irreducible quintic curves

49 types of real singular points for reducible quintic curves

Complete classification verified through symbolic computation

Abstract

There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for reducible real quintic curves. The classification of real singular points for irreducible real quintic curves is originally due to Golubina and Tai. There are 28 types of singular points for irreducible complex quintic curves and 33 types of singular points for reducible complex quintic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, a major component of the proof consists of a sequence of large symbolic computations that can be done nicely using Maple.