Étant donné un ensemble semi-algébrique fermé (non nécessairement compact) de , nous construisons une fonction semi-algébrique positive et de classe telle que et telle que pour suffisamment petit, l’inclusion de dans soit une rétraction. En corollaire, nous obtenons plusieurs formules pour la caractéristique d’Euler de .
Given a closed (not necessarly compact) semi-algebraic set in , we construct a non-negative semi-algebraic function such that and such that for sufficiently small, the inclusion of in is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of .
Classification : 14P10, 14P25
Mots clés : voisinage tubulaire, ensembles semi-algébriques, rétraction, function semi-algébrique approchante quasirégulière, voisinage semi-algébrique approchant quasirégulier
@article{AIF_2009__59_1_429_0, author = {Dutertre, Nicolas}, title = {Semi-algebraic neighborhoods of closed~semi-algebraic sets}, journal = {Annales de l'Institut Fourier}, pages = {429--458}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {1}, year = {2009}, doi = {10.5802/aif.2435}, mrnumber = {2514870}, zbl = {1174.14051}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2435/} }
TY - JOUR AU - Dutertre, Nicolas TI - Semi-algebraic neighborhoods of closed semi-algebraic sets JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 429 EP - 458 VL - 59 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2435/ UR - https://www.ams.org/mathscinet-getitem?mr=2514870 UR - https://zbmath.org/?q=an%3A1174.14051 UR - https://doi.org/10.5802/aif.2435 DO - 10.5802/aif.2435 LA - en ID - AIF_2009__59_1_429_0 ER -
Dutertre, Nicolas. Semi-algebraic neighborhoods of closed semi-algebraic sets. Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 429-458. doi : 10.5802/aif.2435. http://archive.numdam.org/articles/10.5802/aif.2435/
[1] Index of a singular point of a vector field, the Petrovski-Oleinik inequality, and mixed Hodge structures, Funct. Anal. Appl., Volume 12 (1978), pp. 1-14 | MR 498592 | Zbl 0407.57025
[2] Géométrie algébrique réelle, Ergebnisse der Mathematik, 12, Springer-Verlag, 1987 | MR 949442 | Zbl 0633.14016
[3] Integral geometry of tame sets, Geom. Dedicata, Volume 82 (2000), pp. 285-323 | Article | MR 1789065 | Zbl 1023.53057
[4] On the topology of polynomial hypersurfaces, Singularities, Part 1 (Arcata, Calif., 1981), pp.167–178, Proc. Sympos. Pure Math., Volume 40 (1983) | MR 713056 | Zbl 0526.14010
[5] An introduction to o-minimal geometry, in Dottorato di Recerca in Matematica (2000) (Ph. D. Thesis)
[6] An introduction to semi-algebraic geometry, in Dottorato di Recerca in Matematica (2000) (Ph. D. Thesis)
[7] Trivialités en famille, in Real algebraic geometry (Rennes, 1991), pp.193–204, Lecture Notes in Math., 1524, Springer, Berlin, 1992 | MR 1226253 | Zbl 0801.14016
[8] Neighborhoods of algebraic sets, Trans. Amer. Math. Soc., Volume 276 (1983), pp. 517-530 | Article | MR 688959 | Zbl 0529.14013
[9] Geometrical and topological properties of real polynomial fibres, Geom. Dedicata, Volume 105 (2004), pp. 43-59 | Article | MR 2057243 | Zbl 1060.14081
[10] Exposants de Lojasiewicz pour les fonctions semi-algébriques, Ann. Polon. Math., Volume 56 (1992), pp. 123-131 | EuDML 262403 | MR 1159983 | Zbl 0773.14027
[11] A generalized Petrovskii inequality, Funct. Anal. Appl., Volume 8 (1974), pp. 50-56 | Article | MR 350056 | Zbl 0301.14021
[12] A generalized Petrovskii inequality II, Funct. Anal. Appl., Volume 9 (1975), pp. 93-94 | MR 399502 | Zbl 0327.14018
[13] On the local degree of a smooth map, Soobshch. Akad. Nauk Gruz. SSR, Volume 85 (1977), pp. 309-311 | Zbl 0346.55008
[14] Index of a polynomial vector field, Funct. Anal. Appl., Volume 13 (1978), pp. 38-45 | Article | MR 527521 | Zbl 0437.57012
[15] Boundary indices of polynomial -forms with homogeneous components, St. Petersburg Math. J., Volume 10 (1999), pp. 553-575 | MR 1628042 | Zbl 0990.37040
[16] On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier, Volume 48 (1998), pp. 769-783 | Article | EuDML 75302 | Numdam | MR 1644089 | Zbl 0934.32009
[17] Proof of the gradient conjecture of R. Thom, Ann. of Math. (2), Volume 152 (2000), pp. 763-792 | Article | EuDML 124045 | MR 1815701 | Zbl 1053.37008
[18] -stratification of subanalytic functions and the Lojasiewicz inequality, C. R. Acad. Sci. Paris Sér. I Math., Volume 318 (1994), pp. 129-133 | MR 1260324 | Zbl 0799.32007
[19] Une propriété topologique des sous-ensembles analytiques réels, Colloques Internationaux du CNRS, Les équations aux dérivées partielles, éd. B. Malgrange (Paris 1962), 117, Publications du CNRS, Paris, 1963 | MR 160856 | Zbl 0234.57007
[20] Sur les trajectoires du gradient d’une fonction analytique réelle, Seminari di Geometria 1982–1983, Bologna (1984), pp. 115-117 | MR 771152 | Zbl 0606.58045
[21] Milnor fibration at infinity, Indag. Math., Volume 3 (1992), pp. 323-335 | Article | MR 1186741 | Zbl 0806.57021
[22] On trajectories of analytic gradient vector fields, J. Differential Equations, Volume 184 (2002), pp. 215-223 | Article | MR 1929153 | Zbl 1066.58022
[23] On the topology of real algebraic surfaces, Amer. Math. Soc. Transl., 70, Amer. Math. Soc., 1952 | MR 48095
[24] Regularity at infinity of real and complex polynomial functions, Singularity theory (Liverpool, 1996) (London Math. Soc. Lecture Note Ser.), Volume 263, Cambridge Univ. Press, 1999, pp. xx, 249-264 | MR 1709356 | Zbl 0930.58005
[25] Geometric categories and -minimal structures, Duke Math. J., Volume 84 (1996), pp. 497-540 | Article | MR 1404337 | Zbl 0889.03025
Cité par Sources :