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

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 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 d. As an application, we show that for some constants c,β any two isotopic smooth non-singular algebraic compact curves of degree d in R 2 can be connected by an isotopy passing only through algebraic curves of degree exp(cd βd 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 - triangulation of a compact non-singular n-dimensional algebraic hypersurface of degree d.

Soit M n R n+1 une hypersurface compacte lisse. Nous définissons κ(M n ) comme le rapport diam R n+1 (M n )/r(M n )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 )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 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 exp(cd βd 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 d’une hypersurface algébrique de dimension n, non singulière de degré 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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {41},
     number = {1},
     year = {1991},
     pages = {11-25},
     doi = {10.5802/aif.1246},
     zbl = {0746.14022},
     mrnumber = {92j:14070},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1991__41_1_11_0}
}
Nabutovsky, Alexander. Smoothing of real algebraic hypersurfaces by rigid isotopies. Annales de l'Institut Fourier, Volume 41 (1991) no. 1, pp. 11-25. doi : 10.5802/aif.1246. http://www.numdam.org/item/AIF_1991__41_1_11_0/

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