volume under induced metric of preimage set of a regular value under a polynomial map

2010-11-07T19:20:31Z

I am interested in the following kind of polynomial map: $$ f: \mathbb{T}^k \to \mathbb{R}^n$$ where $f$ is a polynomial map of a certain maximum degree $d$, in the sense that if we imbed $\mathbb{T}^k \subset \mathbb{R}^{2k}$ then $f$ is the restriction of some polynomial map. Then my question is, what can one say about the $k-n$-dimensional volume of the set $f^{-1}(p)$ where $p$ is a regular value of $f$, i.e., the differential of $f$ at any preimage of $p$ is surjective onto the tangent space of $\mathbb{R}^n$ at $p$. I would like to get a polynomial bound of this volume in terms of $n,d,k$. There are some papers by Yomdin on zero sets of almost polynomial functions mapped from a unit ball in some euclidean space. But he doesn't seem to mention the word regular and still gets a result about $k-n$ dimensional volume. I am probably missing something. But I couldn't figure it out. Below is the link to his paper. Of course I care about maps in general, not just functions, but I don't need almost polynomial, just polynomial. I feel this should have been known to people like Hassler Whitney, but I am not sure where to find.

http://www.ams.org/proc/1984-090-04/S0002-9939-1984-0733402-5/S0002-9939-1984-0733402-5.pdf

Edit: I just understood why Yomdin didn't have to specify regularity. His function is sufficiently close to a polynomial, and for nondegenerate polynomials, even if the value is not regular, the preimage would still be small, i.e., of strictly lower dimension than the ambient space.

volume under induced metric of preimage set of a regular value under a polynomial map - MathOverflow

volume under induced metric of preimage set of a regular value under a polynomial map