View source for Eisenbud–Levine–Khimshiashvili signature formula

{{Short description|Computes the Poincaré–Hopf index of a real, analytic vector field at a singularity}}
In mathematics, and especially [[Differential topology|differential topology]] and [[Singularity theory|singularity theory]], the '''Eisenbud–Levine–Khimshiashvili signature formula''' gives a way of computing the Poincaré–Hopf index of a real, [[Analytic function|analytic]] [[Vector field|vector field]] at an algebraically isolated singularity.<ref name="arnold et al">{{cite book|first1=Vladimir I.|last1=Arnold|first2=Alexander N.| last2=Varchenko|first3=Sabir M.|last3=Gusein-Zade| title=Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts| translator= Ian Porteous and Mark Reynolds| publisher=Birkhäuser|location= Boston, MA|series= Monographs in Mathematics|volume= 82|year=2009|page=84|isbn=978-0-8176-3187-1|mr=0777682|doi=10.1007/978-1-4612-5154-5}}</ref><ref name="brasselet et al">{{Citation|first1=Jean-Paul|last1=Brasselet| first2=José|last2=Seade|first3=Tatsuo|last3=Suwa|title=Vector fields on singular varieties|pages=123–125|publisher=Springer|location=Berlin|year=2009|isbn=978-3-642-05204-0|mr=2574165|doi=10.1007/978-3-642-05205-7}}</ref>  It is named after [[Biography:David Eisenbud|David Eisenbud]], Harold I. Levine, and George Khimshiashvili.  Intuitively, the index of a vector field near a zero is the number of times the vector field wraps around the sphere.  Because analytic vector fields have a rich algebraic structure, the techniques of [[Commutative algebra|commutative algebra]] can be brought to bear to compute their index.  The signature formula expresses the index of an analytic vector field in terms of the signature of a certain [[Quadratic form|quadratic form]].

== Nomenclature ==
Consider the [[Real coordinate space|''n''-dimensional space]] '''R'''<sup>''n''</sup>. Assume that '''R'''<sup>''n''</sup> has some fixed [[Coordinate system|coordinate system]], and write '''x''' for a point in '''R'''<sup>''n''</sup>, where {{nowrap|1='''x''' = (''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>).}}

Let ''X'' be a [[Vector field|vector field]] on '''R'''<sup>''n''</sup>. For {{nowrap|1=1 ≤ ''k'' ≤ ''n''}} there exist [[Function (mathematics)|functions]] {{nowrap|1=ƒ<sub>''k''</sub> : '''R'''<sup>''n''</sup> → '''R'''}} such that one may express ''X'' as
:<math> X = f_1({\mathbf x})\,\frac{\partial}{\partial x_1} + \cdots + f_n({\mathbf x})\,\frac{\partial}{\partial x_n} . </math>

To say that ''X'' is an ''analytic vector field'' means that each of the functions {{nowrap|1=ƒ<sub>''k''</sub> : '''R'''<sup>''n''</sup> → '''R'''}}  is an [[Analytic function|analytic function]]. One says that ''X'' is ''singular'' at a point '''p''' in '''R'''<sup>''n''</sup> (or that '''p''' is a ''singular point'' of ''X'') if {{nowrap|1=''X''('''p''') = 0}}, i.e. ''X'' vanishes at '''p'''. In terms of the functions {{nowrap|1=ƒ<sub>''k''</sub> : '''R'''<sup>''n''</sup> → '''R'''}} it means that {{nowrap|1=ƒ<sub>''k''</sub>('''p''') = 0}} for all {{nowrap|1=1 ≤ ''k'' ≤ ''n''}}. A singular point '''p''' of ''X'' is called ''isolated'' (or that '''p''' is an ''isolated singularity'' of ''X'') if {{nowrap|1=''X''('''p''') = 0}} and there exists an open neighbourhood {{nowrap|1=''U'' ⊆ '''R'''<sup>''n''</sup>}}, containing '''p''', such that {{nowrap|1=''X''('''q''') ≠ 0}} for all '''q''' in ''U'', different from '''p'''. An isolated singularity of ''X'' is called algebraically isolated if, when considered over the complex domain, it remains isolated.<ref>{{cite journal|last=Arnold|first= Vladimir I.|title=The index of a singular point of a vector field, the Petrovskiĭ-Oleĭnik inequalities, and mixed Hodge structures|journal=Functional Analysis and Its Applications|pages=1–12|volume=12|issue=1|year=1978|doi=10.1007/BF01077558|mr=0498592|s2cid= 123306360}}</ref><ref name="gomez">{{cite journal|last1=Gómex Mont|first1= Xavier |last2=Mardešić|first2= Pavao |title=The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant|journal=[[Annales de l'Institut Fourier]]|issue=47|volume=5|year=1997|pages=1523–1539|url=http://aif.cedram.org/item?id=AIF_1997__47_5_1523_0|mr=1600363}}</ref>

Since the Poincaré–Hopf index ''at a point'' is a purely local invariant (cf. [[Poincaré–Hopf theorem]]), one may restrict one's study to that of [[Germ (mathematics)|germs]]. Assume that each of the ƒ<sub>''k''</sub> from above are ''function germs'', i.e. {{nowrap|1=ƒ<sub>''k''</sub> : ('''R'''<sup>''n''</sup>,0) → ('''R''',0).}} In turn, one may call ''X'' a ''vector field germ''.

== Construction ==
Let ''A''<sub>''n'',0</sub> denote the [[Ring (mathematics)|ring]] of analytic function germs {{nowrap|1=('''R'''<sup>''n''</sup>,0) → ('''R''',0)}}. Assume that ''X'' is a vector field germ of the form
:<math> X = f_1({\mathbf x})\,\frac{\partial}{\partial x_1} + \cdots + f_n({\mathbf x})\,\frac{\partial}{\partial x_n} </math>
with an algebraically isolated singularity at 0. Where, as mentioned above, each of the ƒ<sub>''k''</sub> are function germs {{nowrap|1=('''R'''<sup>''n''</sup>,0) → ('''R''',0)}}. Denote by ''I<sub>X</sub>'' the [[Ideal (ring theory)|ideal]] generated by the ƒ<sub>''k''</sub>, i.e. {{nowrap|1=''I<sub>X</sub>'' = (ƒ<sub>1</sub>, …, ƒ<sub>''n''</sub>).}} Then one considers the local algebra, ''B<sub>X</sub>'', given by the [[Quotient ring|quotient]]
:<math> B_X := A_{n,0} / I_X \, . </math>

The Eisenbud–Levine–Khimshiashvili signature formula states that the index of the vector field ''X'' at 0 is given by the signature of a certain non-degenerate [[Bilinear form|bilinear form]] (to be defined below) on the local algebra ''B<sub>X</sub>''.<ref name="brasselet et al" /><ref name="gomez" /><ref>{{cite journal|last1=Eisenbud|first1= David|last2=Levine| first2= Harold I. | title=An algebraic formula for the degree of a ''C''<sup>∞</sup> map germ|journal=[[Annals of Mathematics]]|volume=106|issue=1|year=1977|pages=19–38|jstor=1971156|doi=10.2307/1971156|mr=0467800}}</ref>

The dimension of <math>B_X</math> is finite if and only if the [[Complexification|complexification]] of ''X'' has an isolated singularity at 0 in '''C'''<sup>''n''</sup>; i.e. ''X'' has an algebraically isolated singularity at 0 in '''R'''<sup>''n''</sup>.<ref name="brasselet et al" /> In this case, ''B<sub>X</sub>'' will be a finite-dimensional, real algebra.

=== Definition of the bilinear form ===
Using the analytic components of ''X'', one defines another analytic germ {{nowrap|1=F : ('''R'''<sup>''n''</sup>,0) → ('''R'''<sup>''n''</sup>,0)}} given by
:<math> F({\mathbf x}) := (f_1({\mathbf x}), \ldots, f_n({\mathbf x})) , </math>
for all {{nowrap|1='''x''' ∈ '''R'''<sup>''n''</sup>}}. Let {{nowrap|1=J<sub>''F''</sub> ∈ ''A''<sub>''n'',0</sub>}} denote the [[Determinant|determinant]] of the Jacobian matrix of ''F'' with respect to the [[Basis (linear algebra)|basis]] {{nowrap|1={∂/∂''x''<sub>1</sub>, …, ∂/∂''x<sub>n</sub>''}.}} Finally, let {{nowrap|1=[J<sub>''F''</sub>] ∈ ''B<sub>X</sub>''}} denote the [[Equivalence class|equivalence class]] of J<sub>''F''</sub>, [[Modular arithmetic|modulo]] ''I<sub>X</sub>''. Using ∗ to denote multiplication in ''B<sub>X</sub>'' one is able to define a non-degenerate bilinear form β as follows:<ref name="brasselet et al" /><ref name="gomez" />
:<math> \beta : B_X \times B_X \stackrel{*}{\longrightarrow} B_X \stackrel{\ell}{\longrightarrow} \R; \ \ \beta(g,h) = \ell(g*h) , </math>

where <math>\scriptstyle \ell</math> is ''any'' linear function such that
:<math> \ell \left( \left[ J_F \right] \right) > 0 . </math>
As mentioned: the signature of β is exactly the index of ''X'' at 0.

== Example ==
Consider the case {{nowrap|1=''n'' = 2}} of a vector field on the plane. Consider the case where ''X'' is given by
:<math>X := (x^3 - 3xy^2) \, \frac{\partial}{\partial x} + (3x^2y - y^3) \, \frac{\partial}{\partial y} . </math>
Clearly ''X'' has an algebraically isolated singularity at 0 since {{nowrap|1=''X'' = 0}} if and only if {{nowrap|1=''x'' = ''y'' = 0.}} The ideal ''I<sub>X</sub>'' is given by {{nowrap|1=(''x''<sup>3</sup> − 3''xy''<sup>2</sup>, 3''x''<sup>2</sup>''y'' − ''y''<sup>3</sup>),}} and

:<math> B_X = A_{2,0} / (x^3 - 3xy^2, 3x^2y - y^3) \cong \R\langle 1, x, y, x^2, xy, y^2, xy^2, y^3, y^4 \rangle . </math>

The first step for finding the non-degenerate, bilinear form β is to calculate the multiplication table of ''B<sub>X</sub>''; reducing each entry modulo ''I<sub>X</sub>''. Whence

{| class="wikitable" style="text-align:center;margin:1em auto;"
!∗
!1
!''x''
!''y''
!''x''<sup>2</sup>
!''xy''
!''y''<sup>2</sup>
!''xy''<sup>2</sup>
!''y''<sup>3</sup>
!''y''<sup>4</sup>
|-
!1
|1
|''x''
|''y''
|''x''<sup>2</sup>
|''xy''
|''y''<sup>2</sup>
|''xy''<sup>2</sup>
|''y''<sup>3</sup>
|''y''<sup>4</sup>
|-
!''x''
|''x''
|''x''<sup>2</sup>
|''xy''
|3''xy''<sup>3</sup>
|''y''<sup>3</sup>/3
|''xy''<sup>2</sup>
|''y''<sup>4</sup>/3
|0
|0
|-
!''y''
|''y''
|''xy''
|''y''<sup>2</sup>
|''y''<sup>3</sup>/3
|''xy''<sup>2</sup>
|''y''<sup>3</sup>
|0
|''y''<sup>4</sup>
|0
|-
!''x''<sup>2</sup>
|''x''<sup>2</sup>
|3''xy''<sup>2</sup>
|''y''<sup>3</sup>/3
|''y''<sup>4</sup>
|0
|''y''<sup>4</sup>/3
|0
|0
|0
|-
!''xy''
|''xy''
|''y''<sup>3</sup>/3
|''xy''<sup>2</sup>
|0
|''y''<sup>4</sup>/3
|0
|0
|0
|0
|-
!''y''<sup>2</sup>
|''y''<sup>2</sup>
|''xy''<sup>2</sup>
|''y''<sup>3</sup>
|''y''<sup>4</sup>/3
|0
|''y''<sup>4</sup>
|0
|0
|0
|-
!''xy''<sup>2</sup>
|''xy''<sup>2</sup>
|''y''<sup>4</sup>/3
|0
|0
|0
|0
|0
|0
|0
|-
!''y''<sup>3</sup>
|''y''<sup>3</sup>
|0
|''y''<sup>4</sup>
|0
|0
|0
|0
|0
|0
|-
!''y''<sup>4</sup>
|''y''<sup>4</sup>
|0
|0
|0
|0
|0
|0
|0
|0
|}
Direct calculation shows that {{nowrap|1=J<sub>''F''</sub> = 9(''x''<sup>4</sup> + 2''x''<sup>2</sup>''y''<sup>2</sup> + ''y''<sup>4</sup>)}}, and so {{nowrap|1=[J<sub>''F''</sub>] = 24''y''<sup>4</sup>.}} Next one assigns values for <math>\scriptstyle \ell</math>. One may take
:<math> \ell(1) = \ell(x) = \ell(y) = \ell(x^2) = \ell(xy) = \ell(y^2) = \ell(xy^2) = \ell(y^3) = 0, \ \text{and} \  \ell(y^4) = 3 .</math>
This choice was made so that <math>\scriptstyle \ell\left( \left[ J_F \right] \right) \, > \, 0</math> as was required by the hypothesis, and to make the calculations involve integers, as opposed to fractions. Applying this to the multiplication table gives the matrix representation of the bilinear form β with respect to the given basis:
<math display="block"> \left[ \begin{array}{ccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 \\
0 & 0 & 0 & 3 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 3 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\
3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array} \right] </math>
The eigenvalues of this matrix are {{nowrap|1=−3, −3, −1, 1, 1, 2, 3, 3 and 4}} There are 3 negative eigenvalues ({{nowrap|1=#''N'' = 3}}), and six positive eigenvalues ({{nowrap|1=#''P'' = 6}}); meaning that the signature of β is {{nowrap|1=#''P'' − #''N'' = 6 − 3 = +3}}. It follows that ''X'' has Poincaré–Hopf index +3 at the origin.

== Topological verification ==
With this particular choice of ''X'' it is possible to verify the Poincaré–Hopf index is +3 by a direct application of the definition of Poincaré–Hopf index.<ref name="milnor">{{Citation|first=John W.|last=Milnor|title=Topology from the Differentiable Viewpoint|publisher=Princeton University Press|year=1997|isbn=978-0-691-04833-8}}</ref> '''This is very rarely the case, and was the reason for the choice of example'''. If one takes polar coordinates on the plane, i.e. {{nowrap|1=''x'' = ''r''&nbsp;cos(θ)}} and {{nowrap|1=''y'' = ''r''&nbsp;sin(θ)}} then {{nowrap|1=''x''<sup>3</sup> − 3''xy''<sup>2</sup> = ''r''<sup>3</sup>cos(3θ)}} and {{nowrap|1=3''x''<sup>2</sup>''y'' − ''y''<sup>3</sup> = ''r''<sup>3</sup>sin(3θ).}} Restrict ''X'' to a circle, centre 0, radius {{nowrap|1= 0 < ε ≪ 1}}, denoted by ''C''<sub>0,ε</sub>; and consider the map {{nowrap|1=''G'' : ''C''<sub>0,ε</sub> → ''C''<sub>0,1</sub>}} given by
:<math> G\colon  X \longmapsto \frac{X}{||X||} .</math>
The Poincaré–Hopf index of ''X'' is, by definition, the topological degree of the map ''G''.<ref name="milnor" /> Restricting ''X'' to the circle ''C''<sub>0,ε</sub>, for arbitrarily small ε, gives
:<math> G(\theta) = (\cos(3\theta),\sin(3\theta)) , \, </math>
meaning that as θ makes one rotation about the circle ''C''<sub>0,ε</sub> in an anti-clockwise direction; the image ''G''(θ) makes three complete, anti-clockwise rotations about the unit circle ''C''<sub>0,1</sub>. Meaning that the topological degree of ''G'' is +3 and that the Poincaré–Hopf index of ''X'' at 0 is +3.<ref name="milnor" />

== References ==
{{reflist}}

{{DEFAULTSORT:Eisenbud-Levine-Khimshiashvili signature formula}}
[[Category:Theorems in differential topology]]
[[Category:Singularity theory]]

{{Sourceattribution|Eisenbud–Levine–Khimshiashvili signature formula}}