The Classification of Real Singularities Using Singular. Part III: Unimodal Singularities of Corank 2

TL;DR

This paper introduces a classification algorithm for real hypersurface singularities of corank 2 and modality 1, enabling the determination of their normal forms and moduli parameters, with implementation in the Singular software.

Contribution

It provides a computational method to classify and identify real singularities of corank 2 and modality 1, including explicit algorithms and software implementation.

Findings

Algorithm successfully classifies singularities into Arnold's normal forms.

Implementation in Singular library realclassify.lib facilitates practical use.

Explicit examples demonstrate the algorithm's effectiveness.

Abstract

We present a classification algorithm for isolated hypersurface singularities of corank 2 and modality 1 over the real numbers. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class by specifying all representatives in Arnold's list of normal forms (Arnold et al. 1985) belonging to this class, and the corresponding values of the moduli parameter. We discuss how to computationally realize the individual steps of the algorithm for all singularities in consideration, and give explicit examples. The algorithm is implemented in the Singular library realclassify.lib.