Let f : Rn → R be a smooth function. We denote by Ω(Rn,R) the infinite-dimensional space of all such functions. Let diff(Rn) denote the infinite-dimensional Lie group of diffeomorphisms Rn → Rn, and diff(R) the infinite-dimensional Lie group of diffeomorphisms R → R. The product group diff(Rn) × diff(R) acts on Ω(Rn,R) in the following way: let φ : Rn → Rn and ψ : R → R be diffeormorphisms and f : Rn → R any smooth function. 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 Rn and a diffeomorphic change of coordinate in R 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 and k ≥ 0 is an integer.
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.