论文标题
光滑的高表情的程度是多少？
What is the Degree of a Smooth Hypersurface?
论文作者
论文摘要
令$ d $为$ \ mathbb {r}^n $和$ f \ in C^{r+2}（d，\ mathbb {r}^k）$中的磁盘。我们处理集合$ j^{r} f^{ - 1}（w）的代数近似问题的问题是$ f $的零集，还是其关键点的集合。 在某些横向条件下，我们证明$ f $可以用多项式地图$ p：d \ to \ mathbb {r}^k $近似，使得相应的奇异性与原始图相关，因此该多项式地图的程度可以由$ c^{r+2 $ f $ f $ f $ f $ f $ f $ f $ f $ f。更准确地说，\ begin {qore} \ text {deg}（p）\ le o \ left（\ frac {\ | f \ | _ { Δ_W）} \ right），\ end {equation} 其中$Δ_W$是一组$ r $ th喷气式扩展名的地图。 $ p $的估算值意味着对奇异性的贝蒂数字进行了估计，但是，使用更精致的工具，我们证明了独立的估计，但仅涉及$ c^{r+1} $ $ f $的数据。 这些结果专门针对C^{2}（d，\ Mathbb {r}）$的零集$ f \的情况，并给出了一种通过代数的近似于公式$ f = 0 $定义的平滑性高表情的方法，该代数具有控制性（由纸张的标题）。特别是，我们表明，紧凑的超曲面$ z \ subset d \ subset \ mathbb {r}^n $，带正触及$ρ（z）> 0 $是零d $中的零设置的多项式$ p $ f lek abenit \ begin \ begin \ begin {equation} \ text} \ text} \ text \ deg}（d p）（p）（p）（p）（p） \ left（1+ \ frac {1} {ρ（z）}+\ frac {5n} {ρ（z）^2} \ right），\ end {qore {equation}，其中$ c（d）> 0 $是持续不断的，取决于磁盘$ d $的大小（尤其是$ z $）的大小。
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, Δ_W)}\right), \end{equation} where $Δ_W$ 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 $ρ(Z)>0$ 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}{ρ(Z)}+\frac{5n}{ρ(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 $Z$).
