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.



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?
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 $\deg F\_i$


