Smoothing of real algebraic hypersurfaces by rigid isotopies

Nabutovsky, Alexander

Annales de l'Institut Fourier, Tome 41 (1991), p. 11-25 / Harvested from Numdam

Résumé

Soit $M_n \subset {\bf R}^{n+1}$ une hypersurface compacte lisse. Nous définissons $\kappa (M^n)$ comme le rapport $\operatorname{diam}_{{\bf R}^{n+1}} (M^n)/r(M^n)$ où $r(M^ n)$ est la distance de $M^n$ à l’ensemble central de $M^n$ (en d’autres termes, $r(M^n)$ est le rayon maximal d’un voisinage tubulaire ouvert de $M^n$ sans self-intersection). Nous prouvons que chaque hypersurface algébrique réelle non-singulière de degré $d$ peut être liée par une isotopie rigide avec une hypersurface algébrique $\Sigma ^n_0$ de degré $d$ telle que $\kappa (\Sigma ^n_0)\le \exp (c(n)d^{\alpha (n)d^{n+1}})$. Ici $c(n)$, $\alpha (n)$ ne dépendent que de $n$, et isotopie rigide est une isotopie qui passe seulement à travers des hypersurfaces algébriques de degré $\le d$.

Comme application de ce résultat, nous démontrons qu’il existe des constantes $c,\beta $ telles que chaque paire de courbes planaires algébriques réelles non-singulières de degré $d$ peut être liée par une isotopie qui passe à travers des courbes algébriques de degré $\le \exp (cd^{\beta d^ 2})$. On en déduit par ailleurs, pour $n$ fixé, une borne supérieure en fonction de $d$, du nombre minimal de simplexes dans une triangulation $C^\infty $ d’une hypersurface algébrique de dimension $n$, non singulière de degré $d$.

Abstract

Define for a smooth compact hypersurface $M^ n$ of ${\bf R}^{n+1}$ its crumpleness $\kappa (M^n)$ as the ratio $\operatorname{diam}_{{\bf R}^{n+1}}(M^ n)/r(M^ n)$, where $r(M^ n)$ is the distance from $M^ n$ to its central set. (In other words, $r(M^ n)$ is the maximal radius of an open non-selfintersecting tube around $M^ n$ in ${\bf R}^{n+1}.)$

We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le \exp (c(n)d^{\alpha (n)d^{n+1}})$. Here $c(n)$, $\alpha (n)$ depend only on $n$, and rigid isotopy means an isotopy passing only through hypersurfaces of degree $\le d$. As an application, we show that for some constants $c,\beta $ any two isotopic smooth non-singular algebraic compact curves of degree $\le d$ in ${\bf R}^ 2$ can be connected by an isotopy passing only through algebraic curves of degree $\le \exp (cd^{\beta d^2})$. As another application, we show how to derive an upper bound in terms of $d$ only (for a fixed $n$) for the minimal number of simplices in a $C^\infty $- triangulation of a compact non-singular $n$-dimensional algebraic hypersurface of degree $d$.

Détail

Publié le : 1991-01-01

MR 92j:14070 | Zbl 0746.14022

DOI : https://doi.org/10.5802/aif.1246

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