# Tubular neighborhood growth of zero set of polynomial of bounded degree in the torus

This question is related to my related post:

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 $p$.

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.

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.

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