Smoothing of real algebraic hypersurfaces by rigid isotopies

Nabutovsky, Alexander. "Smoothing of real algebraic hypersurfaces by rigid isotopies." Annales de l'institut Fourier 41.1 (1991): 11-25. <http://eudml.org/doc/74910>.

@article{Nabutovsky1991,
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 $d$.},
author = {Nabutovsky, Alexander},
journal = {Annales de l'institut Fourier},
keywords = {real algebraic manifolds; crumpleness; rigid isotopy; triangulation},
language = {eng},
number = {1},
pages = {11-25},
publisher = {Association des Annales de l'Institut Fourier},
title = {Smoothing of real algebraic hypersurfaces by rigid isotopies},
url = {http://eudml.org/doc/74910},
volume = {41},
year = {1991},
}

TY - JOUR
AU - Nabutovsky, Alexander
TI - Smoothing of real algebraic hypersurfaces by rigid isotopies
JO - Annales de l'institut Fourier
PY - 1991
PB - Association des Annales de l'Institut Fourier
VL - 41
IS - 1
SP - 11
EP - 25
AB - 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 $d$.
LA - eng
KW - real algebraic manifolds; crumpleness; rigid isotopy; triangulation
UR - http://eudml.org/doc/74910
ER -