In mathematics, and in particular singularity theory, an Ak singularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function.
The notation was introduced by V. I. Arnold.
be a smooth function.
the infinite-dimensional space of all such functions.
denote the infinite-dimensional Lie group of diffeomorphisms
the infinite-dimensional Lie group of diffeomorphisms
The product group
We define the group action as follows: The orbit of f , denoted orb(f), 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 f is said to have a type Ak-singularity if it lies in the orbit of where
By a normal form we mean a particularly simple representative of any given orbit.
The above expressions for f give normal forms for the type Ak-singularities.
The type Ak-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 f. This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish εi = +1 from εi = −1.
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