What is the Degree of a Smooth Hypersurface?

Let $D$ be a disk in $\mathbb{R}^n$ and $f\in C^{r+2}(D, \mathbb{R}^k)$. 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 $W\subset J^{r}(D, \mathbb{R}^k).$ 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 $p:D\to \mathbb{R}^k$ 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 O\left(\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 $f\in C^{2}(D, \mathbb{R})$, 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 $Z\subset D\subset \mathbb{R}^n$ 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 $Z$).