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

  • Janko BöhmEmail author
  • Magdaleen S. Marais
  • Gerhard Pfister
Chapter

Abstract

In Arnold et al. (Singularities of Differential Maps, vol. I. Birkhäuser, Boston, 1985), 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 ≤ 2 and modality ≤ 2. 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.

Keywords

Algorithmic classification Arnold normal forms Hypersurface singularities Moduli parameters 

2010 Mathematics Subject Classification.

Primary 14B05; Secondary 32S25 14Q05 

Notes

Acknowledgement

This research was supported by the Staff Exchange Bursary Programme of the University of Pretoria and DFG SPP 1489.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Janko Böhm
    • 1
    Email author
  • Magdaleen S. Marais
    • 2
  • Gerhard Pfister
    • 1
  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Department of Mathematics and Applied MathematicsUniversity of PretoriaHatfieldSouth Africa

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