Is there a uniform bound on the number of solutions to ${\partial p \over \partial x_i} (x_1,...,x_n) = c_i$ outside a set of measure zero?

Question

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 $(x_1,...,x_n) \in {\mathbb R}^n - Z$ 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.

Answer 1

This follows from basic facts about polynomials, to your generalization of a polynomial map $F: \mathbb R^n \to \mathbb R^n$. Since the Jacobian determinant of $F$ does not vanish identically, its zero set $Z$ (the critical points of $F$) has measure 0, since $F$ is a polynomial. Away from critical values, the inverse image is a dimension 0 manifold, i.e. a finite set. Finally, the size of this finite set is bounded by the product of the degrees of the $F_i$'s by Bezout, since it is the intersection of the $F_i^{-1}(c_i)$, each of which is a hypersurface of degree $\deg F_i$