What is the Degree of a Smooth Hypersurface?

Let $D$ be a disk in $mathbb{R}^n$ and $fin C^{r+2}(D, mathbb{R}^k)$. We deal with the problem of the algebraic approximation of the set $j^{r}f^{-1}(W)$ consisting of the set of points in the disk $D$ where the $r$-th jet extension of $f$ meets a given semialgebraic set $Wsubset J^{r}(D, mathbb{R}^k).$ Examples of sets arising in this way are the zero set of $f$, or the set of its critical points. Under some transversality conditions, we prove that $f$ can be approximated with a polynomial map $p:Dto mathbb{R}^k$ such that the corresponding singularity is diffeomorphic to the original one, and such that the degree of this polynomial map can be controlled by the $C^{r+2}$ data of $f$. More precisely, begin{equation} text{deg}(p)le Oleft(frac{|f|_{C^{r+2}(D, mathbb{R}^k)}}{mathrm{dist}_{C^{r+1}}(f, Delta_W)}right), end{equation} where $Delta_W$ is the set of maps whose $r$-th jet extension is not transverse to $W$. The estimate on the degree of $p$ implies an estimate on the Betti numbers of the singularity, however, using more refined tools, we prove independently a similar estimate, but involving only the $C^{r+1}$ data of $f$. These results specialize to the case of zero sets of $fin C^{2}(D, mathbb{R})$, and give a way to approximate a smooth hypersurface defined by the equation $f=0$ with an algebraic one, with controlled degree (from which the title of the paper). In particular, we show that a compact hypersurface $Zsubset Dsubset mathbb{R}^n$ with positive reach $rho(Z)>0$ is isotopic to the zero set in $D$ of a polynomial $p$ of degree begin{equation} text{deg}(p)leq c(D)cdot 2 left(1+frac{1}{rho(Z)}+frac{5n}{rho(Z)^2}right),end{equation} where $c(D)>0$ is a constant depending on the size of the disk $D$ (and in particular on the diameter of $Z$).