Ak singularity
In mathematics, and in particular singularity theory, an singularity, where is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold.
Let be a smooth function. We denote by the infinite-dimensional space of all such functions. Let denote the infinite-dimensional Lie group of diffeomorphisms and the infinite-dimensional Lie group of diffeomorphisms The product group acts on in the following way: let and be diffeomorphisms and any smooth function. We define the group action as follows:
The orbit of, denoted, of this group action is given by
The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in and a diffeomorphic change of coordinate in such that one member of the orbit is carried to any other. A function is said to have a type -singularity if it lies in the orbit of
where and is an integer.
By a normal form we mean a particularly simple representative of any given orbit. The above expressions for give normal forms for the type -singularities. The type -singularities are special because they are amongst the simple singularities, this means that there are only a finite number of other orbits in a sufficiently small neighbourhood of the orbit of .
This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish from.