# 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?

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.

• What role does $Z$ play (it is mentioned only once)? Jun 15, 2022 at 15:31
• I think they mean "for all $(c_i) \notin Z$" Jun 15, 2022 at 15:45
• I mean the set of $(x_1,...,x_n)$ outside of $Z$, and I've edited the question. I'd prefer it to hold for each $(c_1,...,c_n) \in {\mathbb R}^n$, but if this isn't possible a carefully chosen excluded set of $(c_1,...,c_n)$ might also serve my purposes. Jun 15, 2022 at 15:54

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 $$\deg F_i$$
• Can you provide a reference for saying the size of the finite set is uniformly bounded by the degree of $F$? Also, would this extend to real analytic functions? Jun 15, 2022 at 16:32
• Sorry, actually the simplest thing I can think of bounds it by the product of the degrees, which in your case would be $(n-1)^n$. I can try to think if there's a better bound in your case. It definitely does not extend to real analytic, both bc you need to switch Z to the codomain bc of bump functions, and bc there could be infinitely many preimages (consider sine) Jun 15, 2022 at 17:18