Singular Points of High Multiplicity for Septic Curves

TL;DR

This paper classifies the types of high-multiplicity singular points on real and complex septic algebraic curves, providing a detailed taxonomy and highlighting open problems in the field.

Contribution

It offers a comprehensive classification of singular points of multiplicity four, five, and six on septic curves, including counts for real and complex cases.

Findings

22 types of real singular points of multiplicity six

174 types of real singular points of multiplicity five

At least 182 types of real singular points of multiplicity four

Abstract

For real irreducible algebraic curves of the seventh degree, there are 22 types of singular points of multiplicity six, 174 types of singular points of multiplicity five, and at least 182 types of singular points of multiplicity four. For complex irreducible algebraic curves of the seventh degree, there are 12 types of singular points of multiplicity six, 92 types of singular points of multiplicity five, and at least 92 types of singular points of multiplicity four. In the final section of the paper, a wide variety of open problems on the classification of singular points of plane algebraic cuves is explicitly described.