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

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.

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.

@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, Tome 41 (1991) no. 1, pp. 11-25. doi : 10.5802/aif.1246. http://www.numdam.org/articles/10.5802/aif.1246/

[AMR] R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer, 1988. | Zbl

[ABB] F. Acquistapace, R. Benedetti, F. Broglia, Effectiveness-non effectiveness in semi-algebraic and PL geometry, Inv. Math., 102 (1) (1990), 141-156. | MR | Zbl

[BZ] Yu. Burago, V. Zalgaller, Geometric Inequalities, Springer, 1988. | Zbl

[GPS] J. Goodman, R. Pollack, B. Strumfels, The intrinsic spread of a configuration in Rd, J. Amer. Math. Soc., 3 (1990), 639-651. | Zbl

[GV] D. Grigorjev, N. Vorobjov, Solving systems of polynomial inequalities in subexponential time, J. of Symbolic Computations, 5 (1988), 37-64. | MR | Zbl

[GZK] I.M. Gelfand, A.V. Zelevinsky, M.M. Kapranov, On discriminants of multivariate polynomials, Funct. Analysis and Appl., 24 (1) (1990), 1-4 (in Russian). | Zbl

[L] D. Lazard, Résolutions des systèmes d'équations algébriques, Theor. Comput. Sci., 15 (1981), 77-110. | MR | Zbl

[LF1] V. Lagunov, A. Fet, Extremal problems for hypersurfaces of a given topological type, I, Siberian Math. J., 4(1) (1963), 145-176 (in Russian).

[LF2] V. Lagunov, A. Fet, Extremal problems for hypersurfaces of a given topological type, II, Siberian Math. J., 6(5) (1965), 1026-1036 (in Russian). | Zbl

[M] D. Milman, The central function of the boundary of a domain and its differentiable properties, J. of Geometry, 14 (1980), 182-202. | MR | Zbl

[MW] D. Milman, Z. Waksman, On topological properties of the central set of a bounded domain in Rn, J. of Geometry, 15 (1981), 1-7. | MR | Zbl

[Mo] E.E. Moise, Geometric Topology in Dimensions 2 and 3, Springer, 1977. | MR | Zbl

[N1] A. Nabutovsky, Nonrecursive functions in real algebraic geometry, Bull. Amer. Math. Soc., 20 (1), 61-65. | MR | Zbl

[N2] A. Nabutovsky, Isotopies and nonrecursive functions in real algebraic geometry, in Real Analytic and Algebraic Geometry, edited by M. Galbiati and A. Tognoli, Springer, Lect. Notes in Math., n° 1420, pp. 194-205. | MR | Zbl

[N3] A. Nabutovsky, Number of solutions with a norm bounded by a given constant of a semilinear elliptic PDE with a generic right hand side, to appear in Trans. Amer. Math. Soc. | Zbl

[R] V. Rokhlin, Complex topological characteristics of real algebraic curves, Russian Math. Surveys, 33 (5) (1978), 85-98. | MR | Zbl

[T] J. Thorpe, Elementary Topics in Differential Geometry, Springer, 1979. | MR | Zbl

[VEL] A.G. Vainstein, V.A. Efremovitch, E.A. Loginov, On the skeleton of a Riemann manifold with an edge, Russian Math. Surveys, 33(3) (1978), 181-182. | Zbl

[Vi] O. Viro, Progress in the topology of real algebraic varieties over the last six years, Russian Math. Surveys, 41 (3) (1986), 55-82. | Zbl

[V] N. Vorobjov, Estimates of real roots of a system of algebraic equations, J. of Soviet Math., 34 (1986), 1754-1762. | Zbl

[VW] B.L. Van Der Waerden, Modern Algebra, v. II, Frederik Ungar Publishing Co, 1950.

[Wh] H. Whitney, Geometric Integration Theory, Princeton University Press, Princeton, 1957. | MR | Zbl

Cité par Sources :