The Eisenbud–Levine–Khimshiashvili signature formula

I learned last week of a really cool result, published when I was a first-year undergraduate, that I had not been aware of before.  Maybe everyone knew it except me, but it is so neat I’m going to write about it anyway.

To set the scene, think about the Hopf index theorem for vector fields on a (compact, oriented)  \(n\)-manifold.  Choose a vector field with isolated zeroes.  Associated to each zero is an index – an integer which can be described as the degree of the map \(S^{n-1}\to {\mathbb R}^n\setminus\{0\}\) got by suitably coordinatizing a neighborhood of the critical point and then restricting the vector-field to a little sphere around it.  Hopf showed that the sum of these local indices equals the Euler characteristic.

A generic zero is non-degenerate – the Jacobian of the vector field is non-vanishing – and in that case we know that the index is \(\pm 1\) depending on the sign of the Jacobian (which determines whether the induced map preserves or reverses orientation).  Thus, the index which is defined in terms of the topology of a neighborhood of the zero is in fact determined by the algebraic behavior of the Jacobian at the zero.  At some point, Arnol’d apparently asked whether there was a similar “algebraic” formula for the index in the general (possibly degenerate) case.  There is a beautiful answer to this.  What’s more, the answer is simple enough to explain to an undergraduate class, but the proof seems to be an order of magnitude more difficult.

So what is the answer? Working locally, let’s consider our vector-field to be a map \(f\colon {\mathbb R}^n \to {\mathbb R}^n\), sending 0 to 0.  First we define the local ring \({\mathscr Q}_f\) of \(f\) at 0: this is simply the quotient \( {\mathscr F}/I\) of the ring \(\mathscr F\) of all real-valued functions on \({\mathbb R}^n\) (smooth, or analytic, or polynomial… it doesn’t make much difference) by the ideal generated by the \(n\) components of the vector-valued function \(f\).  The formula applies only in the case where \({\mathscr Q}_f\) is a finite-dimensional real vector space.  For example, consider the standard map of degree 2, \(f(x,y) = (2xy,x^2-y^2)\).  It’s easy to see that \({\mathscr Q}_f\) is 4-dimensional, spanned by the residue classes of 1, \(x\), \(y\), and \(x^2+y^2\).

The Jacobian of \(f\), \(J(f)\), can be considered as an element of \({\mathscr Q}_f\) – in our example \( J(f) = 4(x^2+y^2)\).  As this example suggests, the Jacobian is a nonzero element of this quotient ring.  Choose a linear functional \(\phi\colon {\mathscr Q}_f\to{\mathbb R} \) sending \(J\) to 1 and define a symmetric bilinear form \(B\) by

\[ B(a,b) = \phi(ab). \]

Theorem: This bilinear form is nondegenerate, and is signature is equal to the index of the critical point.

Example: In the example above, choose \(\phi\) to be 1 on \(J\) and zero on the basis elements \(1,x,y\).  The matrix of \(B\) with respect to the standard basis is then

\[ \frac{1}{8}\begin{bmatrix}0 & 0 & 0 & 2\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 2 & 0 & 0 & 0 \end{bmatrix} \]

whose signature is 2 as expected.

Exercise: Show that in the nondegenerate case this recipe does indeed give the sign of the Jacobian at the critical point, as expected.

Arnold, V. I., S. M. Gusein-Zade, and A. N. Varchenko. “The Local Multiplicity of a Holomorphic Map.” In Singularities of Differentiable Maps, Volume 1, 84–114. Modern Birkhäuser Classics. Birkhäuser Boston, 2012.

Eisenbud, David, Harold I. Levine, and Bernard Teissier. “An Algebraic Formula for the Degree of a \(C^\infty\) Map Germ / Sur Une Inégalité à La Minkowski Pour Les Multiplicités.” Annals of Mathematics 106, no. 1 (July 1, 1977): 19–44. doi:10.2307/1971156.

Eisenbud, David. “An Algebraic Approach to the Topological Degree of a Smooth Map.” Bulletin of the American Mathematical Society 84, no. 5 (1978): 751–764.  doi:10.1090/S0002-9904-1978-14509-1.

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