Algorithmic Classification of Real (and Complex) Singularities

In (Arnold et al., 1985), Arnold has obtained normal forms for, in particular, all isolated hypersurface singularities up to modality 1 in the real case, and modality 2 in the complex case. Building on a series of 105 theorems he has developed a classifier that determines a normal form of a given complex singularity up to modality 2. Each normal form is a family of polynomials, which varies with the values of a number of moduli parameters. We call these polynomials normal form equations. A singularity is stably equivalent to at least one, but finitely many, of the normal form equations. Knowing a normal form of a singularity of positive modality does not fix its stable equivalence class. In this talk, we discuss the complete equivalence structure of the unimodal complex- and real singularities that lead to the development of  classification algorithms for real- and complex isolated singularities up to modality 1 and 2, respectively. For a singularity given by a polynomial over the rationals, the algorithms determine its stable equivalence class by specifying all polynomial representatives in Arnold's list of normal forms.

This is joint work with Janko Boehm, Gerhard Pfister and Andreas Steenpass.