Define for a smooth compact hypersurface of its crumpleness as the ratio , where is the distance from to its central set. (In other words, is the maximal radius of an open non-selfintersecting tube around in
We prove that any -dimensional non-singular compact algebraic hypersurface of degree is rigidly isotopic to an algebraic hypersurface of degree and of crumpleness . Here , depend only on , and rigid isotopy means an isotopy passing only through hypersurfaces of degree . As an application, we show that for some constants any two isotopic smooth non-singular algebraic compact curves of degree in can be connected by an isotopy passing only through algebraic curves of degree . As another application, we show how to derive an upper bound in terms of only (for a fixed ) for the minimal number of simplices in a - triangulation of a compact non-singular -dimensional algebraic hypersurface of degree .
Soit une hypersurface compacte lisse. Nous définissons comme le rapport où est la distance de à l’ensemble central de (en d’autres termes, est le rayon maximal d’un voisinage tubulaire ouvert de sans self-intersection). Nous prouvons que chaque hypersurface algébrique réelle non-singulière de degré peut être liée par une isotopie rigide avec une hypersurface algébrique de degré telle que . Ici , ne dépendent que de , et isotopie rigide est une isotopie qui passe seulement à travers des hypersurfaces algébriques de degré .
Comme application de ce résultat, nous démontrons qu’il existe des constantes telles que chaque paire de courbes planaires algébriques réelles non-singulières de degré peut être liée par une isotopie qui passe à travers des courbes algébriques de degré . On en déduit par ailleurs, pour fixé, une borne supérieure en fonction de , du nombre minimal de simplexes dans une triangulation d’une hypersurface algébrique de dimension , non singulière de degré .
@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 = {Institut Fourier}, address = {Grenoble}, volume = {41}, number = {1}, year = {1991}, doi = {10.5802/aif.1246}, mrnumber = {92j:14070}, zbl = {0746.14022}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1246/} }
TY - JOUR AU - Nabutovsky, Alexander TI - Smoothing of real algebraic hypersurfaces by rigid isotopies JO - Annales de l'Institut Fourier PY - 1991 SP - 11 EP - 25 VL - 41 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.1246/ DO - 10.5802/aif.1246 LA - en ID - AIF_1991__41_1_11_0 ER -
%0 Journal Article %A Nabutovsky, Alexander %T Smoothing of real algebraic hypersurfaces by rigid isotopies %J Annales de l'Institut Fourier %D 1991 %P 11-25 %V 41 %N 1 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/aif.1246/ %R 10.5802/aif.1246 %G en %F 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/articles/10.5802/aif.1246/
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