A Classification Algorithm for Complex Singularities of Corank and Modality up to Two

TL;DR

This paper presents a straightforward classification algorithm for isolated hypersurface singularities with corank and modality up to two, building on Arnold's foundational work, and specifies the right equivalence class with explicit polynomial representatives.

Contribution

It introduces a simple algorithm that determines the right equivalence class of singularities up to modality two, extending Arnold's classification to include explicit polynomial representatives.

Findings

Algorithm accurately classifies singularities up to modality 2

Provides explicit polynomial representatives for each class

Builds on Arnold's normal form classification

Abstract

In (Arnold, 1985), V.I. Arnold has obtained normal forms and has developed a classifier for, in particular, all isolated hypersurface singularities over the complex numbers up to modality 2. Building on a series of 105 theorems, this classifier determines the type of the given singularity. However, for positive modality, this does not fix the right equivalence class of the singularity, since the values of the moduli parameters are not specified. In this paper, we present a simple classification algorithm for isolated hypersurface singularities of corank and modality up to two. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class by specifying a polynomial representative in Arnold's list of normal forms.