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 $K$?