ANNALES DE L'INSTITUT FOURIER

Smoothing of real algebraic hypersurfaces by rigid isotopies
Annales de l'Institut Fourier, Tome 41 (1991) no. 1, pp. 11-25.

Soit ${M}_{n}\subset {\mathbf{R}}^{n+1}$ une hypersurface compacte lisse. Nous définissons $\kappa \left({M}^{n}\right)$ comme le rapport ${diam}_{{\mathbf{R}}^{n+1}}\left({M}^{n}\right)/r\left({M}^{n}\right)$$r\left({M}^{n}\right)$ est la distance de ${M}^{n}$ à l’ensemble central de ${M}^{n}$ (en d’autres termes, $r\left({M}^{n}\right)$ 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 ${\Sigma }_{0}^{n}$ de degré $d$ telle que $\kappa \left({\Sigma }_{0}^{n}\right)\le exp\left(c\left(n\right){d}^{\alpha \left(n\right){d}^{n+1}}\right)$. Ici $c\left(n\right)$, $\alpha \left(n\right)$ 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 $\le exp\left(c{d}^{\beta {d}^{2}}\right)$. 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 ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left({M}^{n}\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}\left({M}^{n}\right)/r\left({M}^{n}\right)$, where $r\left({M}^{n}\right)$ is the distance from ${M}^{n}$ to its central set. (In other words, $r\left({M}^{n}\right)$ is the maximal radius of an open non-selfintersecting tube around ${M}^{n}$ in ${\mathbf{R}}^{n+1}.\right)$

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 $\le exp\left(c\left(n\right){d}^{\alpha \left(n\right){d}^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ 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 ${\mathbf{R}}^{2}$ can be connected by an isotopy passing only through algebraic curves of degree $\le exp\left(c{d}^{\beta {d}^{2}}\right)$. 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 $d$.

@article{AIF_1991__41_1_11_0,
author = {Nabutovsky, Alexander},
title = {Smoothing of real algebraic hypersurfaces by rigid isotopies},
journal = {Annales de l'Institut Fourier},
pages = {11--25},
publisher = {Imprimerie Louis-Jean},
volume = {41},
number = {1},
year = {1991},
doi = {10.5802/aif.1246},
zbl = {0746.14022},
mrnumber = {92j:14070},
language = {en},
url = {aif.centre-mersenne.org/item/AIF_1991__41_1_11_0/}
}
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/item/AIF_1991__41_1_11_0/

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