Milnor number

In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. 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. It can be considered both a geometric invariant and an algebraic invariant. This is why it plays an important role in algebraic geometry and singularity theory. Consider a holomorphic complex function germ f: Thus for an n-tuple of complex numbers z 1 , … , z n {displaystyle z_{1},ldots ,z_{n}} we get a complex number f ( z 1 , … , z n ) . {displaystyle f(z_{1},ldots ,z_{n}).} We shall write z := ( z 1 , … , z n ) . {displaystyle z:=(z_{1},ldots ,z_{n}).} We say that f is singular at a point z 0 ∈ C n {displaystyle z_{0}in mathbb {C} ^{n}} if the first order partial derivatives ∂ f / ∂ z 1 , … , ∂ f / ∂ z n {displaystyle partial f/partial z_{1},ldots ,partial f/partial z_{n}} are all zero at z = z 0 {displaystyle z=z_{0}} . As the name might suggest: we say that a singular point z 0 ∈ C n {displaystyle z_{0}in mathbb {C} ^{n}} is isolated if there exists a sufficiently small neighbourhood U ⊂ C n {displaystyle Usubset mathbb {C} ^{n}} of z 0 {displaystyle z_{0}} such that z 0 {displaystyle z_{0}} is the only singular point in U. We say that a point is a degenerate singular point, or that f has a degenerate singularity, at z 0 ∈ C n {displaystyle z_{0}in mathbb {C} ^{n}} if z 0 {displaystyle z_{0}} is a singular point and the Hessian matrix of all second order partial derivatives has zero determinant at z 0 {displaystyle z_{0}} : We assume that f has a degenerate singularity at 0. We can speak about the multiplicity of this degenerate singularity by thinking about how many points are infinitesimally glued. If we now perturb the image of f in a certain stable way the isolated degenerate singularity at 0 will split up into other isolated singularities which are non-degenerate! The number of such isolated non-degenerate singularities will be the number of points that have been infinitesimally glued. Precisely, we take another function germ g which is non-singular at the origin and consider the new function germ h := f + εg where ε is very small. When ε = 0 then h = f. The function h is called the morsification of f. It is very difficult to compute the singularities of h, and indeed it may be computationally impossible. This number of points that have been infinitesimally glued, this local multiplicity of f, is exactly the Milnor number of f. Using some algebraic techniques we can calculate the Milnor number of f effortlessly. By O {displaystyle {mathcal {O}}} denote the ring of function germs ( C n , 0 ) → ( C , 0 ) {displaystyle (mathbb {C} ^{n},0) o (mathbb {C} ,0)} . By J f {displaystyle J_{f}} denote the Jacobian ideal of f: The local algebra of f is then given by the quotient algebra