Question

We know by the standard Implicit Function Theorem that

If $f:\mathbb R^4\rightarrow\mathbb R^2$ is a polynomial (or in fact any continuously differentiable function), then there is a point $a\in\mathbb R^2$ such that $f^{-1}(a)$ is at least two-dimensional.

Now imagine that instead of $\mathbb R^2$ we have an algebraic surface $S\subset\mathbb R^3$, i.e. the zero set of a trivariate polynomial. It's reasonable that the statement still holds. A general statement along these lines would be something like

If $A,B$ are two algebraic sets such that $\dim(A)>\dim(B)$, and if $f:A\rightarrow B$ is a polynomial mapping, then there is a point $b\in B$ such that $f^{-1}(b)$ is of dimension at least $\dim(A)-\dim(B)$.

Is it a well-known theorem? Any reference for it?

Answer 1

This is a well-known theorem. I imagine it has been routinely used for many years, so tracking down a historical reference would be hard. The following argument will work for any $f$ which is definable over the real field, i.e. whose graph is a semi-algebraic set.

The graph $\Gamma \subset A\times B$ of your mapping has dimension $\dim(A)$, since the restriction to $\Gamma$ of the projection $\pi_A: A\times B \to A$ is 1-to-1. Now do a cylindrical algebraic decomposition of $A\times B$ adapted to $\Gamma$ and the projection $\pi_B$. Then, any cell in $f(B)$ is the image of a cell of $\Gamma$. In particular, this is true for all the $\dim(A)$-dimensional cells that appear in $\Gamma$; if $C$ is such a cell, any point $b$ in the image $\pi_B(C)$ verifies $\dim(f^{-1}(b)) \geq \dim(A)-\dim(\pi_B(C)) \geq \dim(A)-\dim(B)$.

So not only can you find such a $b$, but there should be quite a few of them.

Answer 2

I think Qing Liu's "Algebraic geometry and algebraic curves" (especially 4.3) answers your question (and gives appropriate conditions).