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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.

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.

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I believe this paper should help you – rpotrie Nov 8 '10 at 17:24
Thanks rpotrie! Do you happen to know any paper that gives more explicit bounds? All I need is a polynomial bound in terms of the relevant quantities. I feel some integral geometric method could be helpful here. – John Jiang Nov 8 '10 at 17:33

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