Smoothing of real algebraic hypersurfaces by rigid isotopies

Nabutovsky, Alexander

doi:10.5802/aif.1246

Nabutovsky, Alexander

Annales de l'Institut Fourier, Tome 41 (1991) no. 1, pp. 11-25.

Résumé
Abstract

Soit $M_{n} \subset R^{n + 1}$ une hypersurface compacte lisse. Nous définissons $κ (M^{n})$ comme le rapport ${diam}_{R^{n + 1}} (M^{n}) / r (M^{n})$ 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 $Σ_{0}^{n}$ de degré $d$ telle que $κ (Σ_{0}^{n}) \leq exp (c (n) d^{α (n) d^{n + 1}})$ . Ici $c (n)$ , $α (n)$ ne dépendent que de $n$ , et isotopie rigide est une isotopie qui passe seulement à travers des hypersurfaces algébriques de degré $\leq d$ .

Comme application de ce résultat, nous démontrons qu’il existe des constantes $c, β$ 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 $\leq exp (c d^{β d^{2}})$ . 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 $d$ .

Define for a smooth compact hypersurface $M^{n}$ of $R^{n + 1}$ its crumpleness $κ (M^{n})$ as the ratio ${diam}_{R^{n + 1}} (M^{n}) / r (M^{n})$ , 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 $R^{n + 1} .)$

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 $\leq exp (c (n) d^{α (n) d^{n + 1}})$ . Here $c (n)$ , $α (n)$ depend only on $n$ , and rigid isotopy means an isotopy passing only through hypersurfaces of degree $\leq d$ . As an application, we show that for some constants $c, β$ any two isotopic smooth non-singular algebraic compact curves of degree $\leq d$ in $R^{2}$ can be connected by an isotopy passing only through algebraic curves of degree $\leq exp (c d^{β d^{2}})$ . 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 $d$ .

MR Zbl

DOI : 10.5802/aif.1246

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     author = {Nabutovsky, Alexander},
     title = {Smoothing of real algebraic hypersurfaces by rigid isotopies},
     journal = {Annales de l'Institut Fourier},
     pages = {11--25},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {41},
     number = {1},
     year = {1991},
     doi = {10.5802/aif.1246},
     zbl = {0746.14022},
     mrnumber = {92j:14070},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1246/}
}

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Nabutovsky, Alexander. Smoothing of real algebraic hypersurfaces by rigid isotopies. Annales de l'Institut Fourier, Tome 41 (1991) no. 1, pp. 11-25. doi : 10.5802/aif.1246. https://aif.centre-mersenne.org/articles/10.5802/aif.1246/

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