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. 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 \R \to \R. 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 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 \R^n 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 .