volume under induced metric of preimage set of a regular value under a polynomial map - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:50:38Z http://mathoverflow.net/feeds/question/45199 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45199/volume-under-induced-metric-of-preimage-set-of-a-regular-value-under-a-polynomial volume under induced metric of preimage set of a regular value under a polynomial map John Jiang 2010-11-07T19:20:31Z 2010-11-07T19:30:48Z <p>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.</p> <p><a href="http://www.ams.org/proc/1984-090-04/S0002-9939-1984-0733402-5/S0002-9939-1984-0733402-5.pdf" rel="nofollow">http://www.ams.org/proc/1984-090-04/S0002-9939-1984-0733402-5/S0002-9939-1984-0733402-5.pdf</a></p> <p>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.</p>