## Tbilisi Mathematical Journal

- Tbilisi Math. J.
- Volume 2 (2009), 95-134.

### Singular points of real quartic and quintic curves

David A. Weinberg and Nicholas J. Willis

#### Abstract

There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D. A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. 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, the proof consists of a sequence of large symbolic computations that can be done nicely using Maple.

#### Note

The authors wish to thank Tomas Recio (Universidad de Cantabria, Santander, Spain), Carlos Andradas (Universidad Complutense de Madrid, Spain), Eugenii Shustin (Tel-Aviv University), Jeffrey M. Lee (Texas Tech University) and Anatoly Korchagin (Texas Tech University) for several useful conversations. We also wish to thank the referees for several suggestions that greatly helped to improve the clarity in this paper.

#### Article information

**Source**

Tbilisi Math. J., Volume 2 (2009), 95-134.

**Dates**

Received: 30 June 2008

Revised: 3 December 2009

Accepted: 24 December 2009

First available in Project Euclid: 12 June 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.tbilisi/1528768844

**Mathematical Reviews number (MathSciNet)**

MR2673506

**Zentralblatt MATH identifier**

1200.14111

**Subjects**

Primary: 14P25: Topology of real algebraic varieties

Secondary: 14H20: Singularities, local rings [See also 13Hxx, 14B05] 14B99: None of the above, but in this section

**Keywords**

Real quartic curves Real quintic curves

#### Citation

Weinberg, David A.; Willis, Nicholas J. Singular points of real quartic and quintic curves. Tbilisi Math. J. 2 (2009), 95--134. https://projecteuclid.org/euclid.tbilisi/1528768844