Classification of Complex Singularities with Non-Degenerate Newton Boundary

TL;DR

This paper extends Arnold's classification of hypersurface singularities to a broader class with non-degenerate Newton boundaries, providing an algorithmic normal form classifier for singularities with corank ≤ 2.

Contribution

It develops a new normal form theorem and an algorithmic classifier for singularities with non-degenerate Newton boundaries, unbounded by modality or Milnor number.

Findings

Algorithmic classifier determines normal forms for a large class of singularities.

Proved a normal form theorem for germs with non-degenerate Newton boundary.

Implemented algorithms in Singular library arnold.lib.

Abstract

In his groundbreaking work on classification of singularities with regard to right and stable equivalence of germs, Arnold has listed normal forms for all isolated hypersurface singularities over the complex numbers with either modality less than or equal to two or Milnor number less than or equal to 16. Moreover, he has described an algorithmic classifier, which determines the type of a given such singularity. In the present paper, we extend Arnold's work to a large class of singularities which is unbounded with regard to modality and Milnor number. We develop an algorithmic classifier, which determines a normal form for any singularity with corank less than or equal to two which is equivalent to a germ with non-degenerate Newton boundary in the sense of Kouchnirenko. In order to realize the classifier, we prove a normal form theorem: Suppose K is a mu-constant stratum of the jet space which contains a germ with a non-degenerate Newton boundary. We first observe that all germs in K are equivalent to some germ with the same fixed non-degenerate Newton boundary. We then prove that all right-equivalence classes of germs in K can be covered by a single normal form obtained from a regular basis of an appropriately chosen special fiber. All algorithms are implemented in the library arnold.lib for the computer algebra system Singular.