Tubular neighborhood growth of zero set of polynomial of bounded degree in the torus - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:35:22Z http://mathoverflow.net/feeds/question/45115 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45115/tubular-neighborhood-growth-of-zero-set-of-polynomial-of-bounded-degree-in-the-to Tubular neighborhood growth of zero set of polynomial of bounded degree in the torus John Jiang 2010-11-06T23:08:32Z 2010-11-07T01:03:29Z <p>This question is related to my related post:</p> <p><a href="http://mathoverflow.net/questions/45005/volume-growth-of-tubular-neigbhorhood-of-critical-values-of-an-algebraic-differen" rel="nofollow">Volume growth of tubular neigbhorhood of critical values of an algebraic/differentiable map</a> </p> <p>The setting here is as follows: Let $p: \mathbb{R}^{2k} \to \mathbb{R}$ be a polynomial of maximum degree $d$. What can one say about the volume growth of the tubular neighborhood $T_\epsilon(p^{-1}(0)) \cap \mathbb{T}^k$ in $\mathbb{T}^k$, where $\mathbb{T}^k$ is imbedded in $\mathbb{R}^{2k}$ in the obvious way? I.e., each two dimensions of $\mathbb{R}^{2k}$ accommodates one copy of $\mathbb{T}^1$. In some sense, I want to understand the $k-1$-dimensional volume of the zero set of $p$. </p> <p>Note the bound on volume growth does not have to be sharp at all. If one can show it's polynomial in $d$ I am already very happy. </p> <p>So I found a paper by Hassler Whitney on elementary structures of real algebraic varieties, as well as some estimates on metric entropy of critical values of algebraic maps by Yomdin.</p>