Suppose $p(x_1,...,x_n)$ is a polynomial in $n$ real variables whose Hessian is not identically zero. Can we say that there is a set $Z \subset {\mathbb R}^n$ of measure zero and a constant $M$ such that for all real numbers $c_1,...,c_n$ the cardinality of the set of $(x_1,...,x_n)$ for which ${\partial p \over \partial x_i}(x_1,...,x_n) = c_i$ for each $i$ is at most $M$? It's possible I might be able to exclude some $(c_1,...,c_n)$ but I'd have be careful what I was excluding.

I've looked around online at resources on systems of polynomial equations and nothing really applied. Also, the above might generalize to systems $p_i(x_1,...,x_n) = c_i$ for $(p_1,...,p_n)$ whose Jacobian is not identically zero, but I only need it in the above situation.

Basically, if this follows from something well-known, or someone has pointers on where to look I'd be appreciative.