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In mathematics, and particularly [[singularity theory]], the '''Milnor number''', named after [[John Milnor]], is an invariant of a function germ.
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If ''f'' is a complex-valued holomorphic [[germ (mathematics)|function germ]] then the Milnor number of ''f'', denoted ''μ(f)'', is either an [[integer]] greater than or equal to [[zero]], or it is [[Infinity|infinite]]. It can be considered both a [[differential geometry|geometric]] [[Invariant (mathematics)|invariant]] and an [[abstract algebra|algebraic]] invariant. This is why it plays an important role in [[algebraic geometry]] and [[singularity theory]].
 
== Geometric interpretation ==
Consider a holomorphic [[complex numbers|complex]] [[germ (mathematics)|function germ]] ''f'':
:<math> f : (\mathbb{C}^n,0) \to (\mathbb{C},0) \ . </math>
Thus for an ''n''-tuple of [[complex numbers]] <math>z_1,\ldots,z_n</math> we get a [[complex number]] <math>f(z_1,\ldots,z_n).</math> We shall write <math>z := (z_1,\ldots,z_n).</math>
 
We say that ''f'' is singular at a point <math>z_0 \in \mathbb{C}^n</math> if the first order [[partial derivatives]] <math>\partial f / \partial z_1, \ldots, \partial f / \partial z_n</math> are all zero at <math>z = z_0</math>. As the name might suggest: we say that a singular point <math>z_0 \in \mathbb{C}^n</math> is isolated if there exists a sufficiently small [[neighbourhood (mathematics)|neighbourhood]] <math>U \subset \mathbb{C}^n</math> of <math>z_0</math> such that <math>z_0</math> 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 <math>z_0 \in \mathbb{C}^n</math> if <math>z_0</math> is a singular point and the [[Hessian matrix]] of all second order partial derivatives has zero [[determinant]] at <math>z_0</math>:
:<math>  \det\left( \frac{\partial^2 f}{\partial z_i \partial z_j} \right)_{1 \le i \le j \le n}^{z = z_0} =0. </math>
 
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 [[Perturbation theory|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''.
 
== Algebraic interpretation ==
Using some [[abstract algebra|algebraic]] techniques we can calculate the Milnor number of ''f'' effortlessly. By <math>\mathcal{O}</math> denote the [[ring (mathematics)|ring]] of function germs <math>(\mathbb{C}^n,0) \to (\mathbb{C},0)</math>. By <math>J_f</math> denote the [[Jacobian ideal]] of ''f'':
:<math> J_f := \left\langle \frac{\partial f}{\partial z_i} : 1 \le i \le n \right\rangle. </math>
The local algebra of ''f'' is then given by the [[Quotient space (linear algebra)|quotient]] [[algebra]]
:<math> \mathcal{A}_f := \mathcal{O} / J_f. </math>
Notice that this quotient space will actually be a [[vector space]], although it may not be finite dimensional. The Milnor number is then equal to the complex dimension of the local algebra:
:<math> \mu(f) = \dim_{\mathbb{C}} \mathcal{A}_f \ . </math>
It follows from Hilbert's [[Nullstellensatz]] that <math>\mu(f)</math> is finite if and only if the origin is an ''isolated'' critical point of ''f''; that is, there is a neighbourhood of 0 in <math>\mathbb{C}^n</math> such that the only critical point of ''f'' inside that neighbourhood is at 0.
 
== Examples ==
Here we give some worked examples in two variables. Working with only one is too simple and does not give a feel for the techniques, whereas working with three variables can be quite tricky. Two is a nice number. Also we stick to polynomials. If ''f'' is only [[holomorphic]] and not a polynomial, then we could have worked with the [[power series]] expansion of ''f''.
=== 1 ===
 
 
Consider a function germ with a non-degenerate singularity at ''0'', say <math>f(x,y) = x^2 + y^2</math>. The Jacobian ideal is just <math> \langle 2x, 2y \rangle = \langle x, y \rangle </math>. We next compute the local algebra:
:<math> \mathcal{A}_f = \mathcal{O} / \langle x, y \rangle = \langle 1 \rangle . </math>
To see why this is true we can use [[Hadamard's lemma]] which says that we can write any function <math>h\in\mathcal{O}</math> as
:<math> h(x,y) = k + xh_1(x,y) + yh_2(x,y) </math>
for some constant ''k'' and functions <math>h_1</math> and <math>h_2</math> in <math>\mathcal{O}</math> (where either <math>h_1</math> or <math>h_2</math> or both may be exactly zero). So, modulo functional multiples of ''x'' and ''y'', we can write ''h'' as a constant. The space of constant functions is spanned by 1, hence  <math>\mathcal{A}_f = \langle 1 \rangle</math>
 
It follows that ''μ(f) = 1''. It is easy to check that for any function germ ''g'' with a non-degenerate singularity at ''0'' we get ''μ(g) = 1''.
 
Note that applying this method to a non-singular function germ ''g'' we get ''μ(g) = 0''.
 
=== 2 ===
 
 
Let <math>f(x,y) = x^3 + xy^2</math>, then
:<math> \mathcal{A}_f = \mathcal{O} / \langle 3x^2 - y^2, xy \rangle = \langle 1, x, y, x^2 \rangle . </math>
So in this case <math>\mu(f) = 4</math>.
 
=== 3 ===
 
One can show that if <math>f(x,y) = x^2y^2 + y^3</math> then <math>\mu(f) = \infty.</math>
 
This can be ''explained'' by the fact that ''f'' is singular at every point of the ''x''-axis.
 
== Versal Deformations ==
Let ''f'' have finite Milnor number μ, and let <math>g_1,\ldots, g_{\mu}</math> be a [[Basis (linear algebra)|basis]] for the local algebra, considered as a vector space. Then a miniversal deformation of ''f'' is given by
:<math> F : (\mathbb{C}^n \times \mathbb{C}^{\mu},0) \to (\mathbb{C},0) ,</math>
:<math> F(z,a) := f(z) + a_1g_1(z) + \cdots + a_{\mu}g_{\mu}(z) ,</math>
where <math>(a_1,\dots,a_{\mu})\in \mathbb{C}^{\mu}</math>.
These deformations (or [[unfolding (functions)|unfolding]]s) are of great interest in much of science.
 
==Invariance==
We can collect function germs together to construct [[equivalence class]]es. One standard equivalence is [[A-equivalence|''A''-equivalence]]. We say that two function germs <math>f,g : (\mathbb{C}^n,0) \to (\mathbb{C},0)</math> are ''A''-equivalent if there exist [[diffeomorphism]] germs <math> \phi : (\mathbb{C}^n,0) \to (\mathbb{C}^n,0)</math> and <math>\psi : (\mathbb{C},0) \to (\mathbb{C},0)</math> such that <math>f \circ \phi = \psi \circ g</math>: there exists a diffeomorphic change of variable in both [[domain of a function|domain]] and [[range of a function|range]] which takes ''f'' to ''g''.
 
The Milnor number does not offer a complete invariant for function germs. We do have that if ''f'' and ''g'' are ''A''-equivalent then ''μ(f) = μ(g)''.
The converse is false: there exist function germs ''f'' and ''g'' with ''μ(f) = μ(g)'' which are not ''A''-equivalent. To see this consider <math>f(x,y) = x^3+y^3</math> and <math>g(x,y) = x^2+y^5</math>. We have <math>\mu(f) = \mu(g) = 4</math> but ''f'' and ''g'' are clearly not ''A''-equivalent since the [[Hessian matrix]] of ''f'' is equal to zero while that of ''g'' is not (and the rank of the Hessian is an ''A''-invariant, as is easy to see).
 
== References ==
* {{Cite book | authorlink=Vladimir Arnold | last1=Arnold | first1=V.I. | last2=Gussein-Zade | first2=S.M. | last3=Varchenko | first3=A.N. | title=Singularities of differentiable maps | volume=Volume 1 | year=1985 | publisher=Birkhauser}}
* {{cite book | last=Gibson | first=Christopher G. | title=Singular Points of Smooth Mappings | series=Research Notes in Mathematics | year=1979 | publisher=Pitman}}
* {{cite book | last=Milnor | first=John | authorlink=John Milnor | title=Morse Theory | series=Annals of Mathematics Studies | year=1963 | publisher=Princeton University Press}}
* {{cite book | last=Milnor | first=John | authorlink=John Milnor | title=Singular points of Complex Hypersurfaces | series=Annals of Mathematics Studies | year=1969 | publisher=Princeton University Press}}
 
{{DEFAULTSORT:Milnor Number}}
[[Category:Singularity theory]]
[[Category:Algebraic geometry]]

Latest revision as of 17:18, 14 March 2014

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