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Suppose I have a map $f: \mathbb R^{N} \mapsto \mathbb R^{N}$ of multivariate polynomial form of degree $K$:

$ f^i: X \mapsto A^{i}_0 + A^{ij}_1 X^{j} + A^{ijk}_2 X^j X^k + \ldots + A^{i i_1 \cdots i_K}_K X^{i_1} \cdots X^{i_K} $

(a sum is implied for each repeated index).

What can be said about the topology of the manifold defined by

$ J(f) = \det \left( \frac{\partial f^i}{\partial X^j } \right) = 0 .$ For instance, what can be said of its dimensionality? How does it depend on $N$ and $K$?

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    @3Sphere: I think you're referring to what's called the Einstein summation convention. lurscher's convention is of his own choosing and he's not bound by any convention of Einstein.2011-11-24

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The only thing special about your variety is that it's the zero-set of the determinant of a matrix such that the mixed partials (of the matrix) agree. So in dimension 1, there's nothing special at all -- every variety is of this form. In higher dimensions, take any polynomials $a_{ij} \in \mathbb R[x_1,\cdots,x_n]$ such that $\frac{\partial a_{ij}}{\partial x_k} = \frac{\partial a_{ik}}{\partial x_j}$ for all $\{i,j,k\}$. Then the variety $Det\begin{pmatrix}a_{11} & \cdots & a_{1n} \\ \cdots & & \cdots \\ a_{n1} & \cdots & a_{nn} \end{pmatrix}=0$ is of the form you're interested in, since the matrix is the Jacobian of some function.

I suppose there's perhaps something you could say about such varieties but it seems like you can get all kinds of behaviour.

Example: $Det\begin{pmatrix}x+x^2 & y-y^{100} \\ x+y & x+y^2\end{pmatrix} = x^3+(y^2+1)x^2 + (-y^{101}+y^2)x-y^{102}$

and so on. Do you have a reason to think there's anything special about these varieties?

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    @lurscher: I think perhaps you're thinking about your Taylor series convergence issue in a perhaps not too useful way. For example, $f(x) = 1-x^2$ has a Taylor expansion which converges with infinite radius. But the derivative matrix is singular at $x=0$.2011-11-24