In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ.
If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ(f), is either a nonnegative integer, or is infinite.
This is why it plays an important role in algebraic geometry and singularity theory.
Consider a holomorphic complex function germ and denote by
In particular, the multiplicity of the gradient is finite by an application of Rückert's Nullstellensatz.
Note that the multiplicity of the gradient is finite if and only if the origin is an isolated critical point of f. Milnor originally[1] introduced
are nonsingular manifolds of real dimension
It is also diffeomorphic to the fiber of the Milnor fibration map.
is equal to the Milnor number and it has homology of a point in dimension less than
For example, a complex plane curve near every singular point
circles (Milnor number is a local property, so it can have different values at different singular points).
Thus the following equalities hold: Another way of looking at Milnor number is by perturbation.
is a singular point and the Hessian matrix of all second order partial derivatives has zero determinant at
The multiplicity of this degenerate singularity may be considered by thinking about how many points are infinitesimally glued.
If the image of f is now perturbed in a certain stable way the isolated degenerate singularity at 0 will split up into other isolated singularities which are non-degenerate.
Precisely, another function germ g which is non-singular at the origin is taken and considered the new function germ h := f + εg where ε is very small.
This number of points that have been infinitesimally glued, this local multiplicity of f, is exactly the Milnor number of f. Further contributions[2] give meaning to Milnor number in terms of dimension of the space of versal deformations, i.e. the Milnor number is the minimal dimension of parameter space of deformations that carry all information about initial singularity.
Given below are some worked examples of polynomials in two variables.
Working with only a single variable is too simple and does not give an appropriate illustration of the techniques, whereas working with three variables can be cumbersome.
Note that if f is only holomorphic and not a polynomial, then the power series expansion of f can be used.
Consider a function germ with a non-degenerate singularity at 0, say
Computing the local algebra: Hadamard's lemma, which says that any function
It is easy to check that for any function germ g with a non-degenerate singularity at 0, μ(g) = 1.
Note that applying this method to a non-singular function germ g yields μ(g) = 0.
This can be explained by the fact that f is singular at every point of the x-axis.
be a basis for the local algebra, considered as a vector space.
These deformations (or unfoldings) are of great interest in much of science.
[citation needed] Function germs can be collected together to construct equivalence classes.
: there exists a diffeomorphic change of variable in both domain and range which takes f to g. If f and g are A-equivalent then μ(f) = μ(g).
[citation needed] Nevertheless, the Milnor number does not offer a complete invariant for function germs, i.e. the converse is false: there exist function germs f and g with μ(f) = μ(g) which are not A-equivalent.