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.
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Notes
- 1.
We say that f is (semi-)quasihomogeneous if there exists a weight w such that f is (semi-)quasihomogeneous with respect to w.
- 2.
We say that the singularity defined by f has non-degenerate Newton boundary if there exists a germ \(\tilde{f} \in \mathbb{C}[[x,y]]\) with \(f \sim \tilde{ f}\) which has non-degenerate Newton boundary. We use the analogous terminology also for semi-quasihomogeneous.
References
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Acknowledgement
This research was supported by the Staff Exchange Bursary Programme of the University of Pretoria and DFG SPP 1489.
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Böhm, J., Marais, M.S., Pfister, G. (2017). A Classification Algorithm for Complex Singularities of Corank and Modality up to Two. In: Decker, W., Pfister, G., Schulze, M. (eds) Singularities and Computer Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-28829-1_2
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DOI: https://doi.org/10.1007/978-3-319-28829-1_2
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