Semi-algebraic neighborhoods of closed semi-algebraic sets
Annales de l'Institut Fourier, Volume 59 (2009) no. 1, p. 429-458

Given a closed (not necessarly compact) semi-algebraic set X in n , we construct a non-negative semi-algebraic 𝒞 2 function f such that X=f -1 (0) and such that for δ>0 sufficiently small, the inclusion of X in f -1 ([0,δ]) is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of X.

Étant donné un ensemble semi-algébrique fermé (non nécessairement compact) X de n , nous construisons une fonction semi-algébrique f positive et de classe 𝒞 2 telle que X=f -1 (0) et telle que pour δ>0 suffisamment petit, l’inclusion de X dans f -1 ([0,δ]) soit une rétraction. En corollaire, nous obtenons plusieurs formules pour la caractéristique d’Euler de X.

DOI : https://doi.org/10.5802/aif.2435
Classification:  14P10,  14P25
Keywords: Tubular neighborhood, semi-algebraic sets, retraction, quasiregular approaching semi-algebraic function, quasiregular approaching semi-algebraic neighborhood
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     pages = {429-458},
     doi = {10.5802/aif.2435},
     mrnumber = {2514870},
     zbl = {1174.14051},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_1_429_0}
}
Dutertre, Nicolas. Semi-algebraic neighborhoods of closed semi-algebraic sets. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 429-458. doi : 10.5802/aif.2435. http://www.numdam.org/item/AIF_2009__59_1_429_0/

[1] Arnold, V. I. Index of a singular point of a vector field, the Petrovski-Oleinik inequality, and mixed Hodge structures, Funct. Anal. Appl., Tome 12 (1978), pp. 1-14 | MR 498592 | Zbl 0407.57025

[2] Bochnak, J.; Coste, M.; Roy, M. F. Géométrie algébrique réelle, Springer-Verlag, Ergebnisse der Mathematik, Tome 12 (1987) | MR 949442 | Zbl 0633.14016

[3] Broecker, L.; Kuppe, M. Integral geometry of tame sets, Geom. Dedicata, Tome 82 (2000), pp. 285-323 | Article | MR 1789065 | Zbl 1023.53057

[4] Broughton, S. A. On the topology of polynomial hypersurfaces, Singularities, Part 1 (Arcata, Calif., 1981), pp.167–178, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, Tome 40 (1983) | MR 713056 | Zbl 0526.14010

[5] Coste, M. An introduction to o-minimal geometry, in Dottorato di Recerca in Matematica, Dip. Mat. Univ. Pisa. Istituti Editoriali e Poligrafici Internazionali, Pisa (2000) (Ph. D. Thesis)

[6] Coste, M An introduction to semi-algebraic geometry, in Dottorato di Recerca in Matematica, Dip. Mat. Univ. Pisa. Istituti Editoriali e Poligrafici Internazionali, Pisa (2000) (Ph. D. Thesis)

[7] Coste, M.; Reguiat, M. Trivialités en famille, in Real algebraic geometry (Rennes, 1991), pp.193–204, Springer, Berlin, Lecture Notes in Math., Tome 1524 (1992) | MR 1226253 | Zbl 0801.14016

[8] Durfee, A. H. Neighborhoods of algebraic sets, Trans. Amer. Math. Soc., Tome 276 (1983), pp. 517-530 | Article | MR 688959 | Zbl 0529.14013

[9] Dutertre, N. Geometrical and topological properties of real polynomial fibres, Geom. Dedicata, Tome 105 (2004), pp. 43-59 | Article | MR 2057243 | Zbl 1060.14081

[10] Fekak, A. Exposants de Lojasiewicz pour les fonctions semi-algébriques, Ann. Polon. Math., Tome 56 (1992), pp. 123-131 | MR 1159983 | Zbl 0773.14027

[11] Kharlamov, V. M. A generalized Petrovskii inequality, Funct. Anal. Appl., Tome 8 (1974), pp. 50-56 | Article | MR 350056 | Zbl 0301.14021

[12] Kharlamov, V. M. A generalized Petrovskii inequality II, Funct. Anal. Appl., Tome 9 (1975), p. 93-94 | MR 399502 | Zbl 0327.14018

[13] Khimshiashvili, G. M. On the local degree of a smooth map, Soobshch. Akad. Nauk Gruz. SSR, Tome 85 (1977), pp. 309-311 | Zbl 0346.55008

[14] Khovanskii, A. G. Index of a polynomial vector field, Funct. Anal. Appl., Tome 13 (1978), pp. 38-45 | Article | MR 527521 | Zbl 0437.57012

[15] Khovanskii, A. G. Boundary indices of polynomial 1-forms with homogeneous components, St. Petersburg Math. J., Tome 10 (1999), pp. 553-575 | MR 1628042 | Zbl 0990.37040

[16] Kurdyka, K. On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier, Tome 48 (1998), pp. 769-783 | Article | Numdam | MR 1644089 | Zbl 0934.32009

[17] Kurdyka, K.; Mostowski, T.; Parusinski, A. Proof of the gradient conjecture of R. Thom, Ann. of Math. (2), Tome 152 (2000), pp. 763-792 | Article | MR 1815701 | Zbl 1053.37008

[18] Kurdyka, K.; Parusinski, A. w f -stratification of subanalytic functions and the Lojasiewicz inequality, C. R. Acad. Sci. Paris Sér. I Math., Tome 318 (1994), pp. 129-133 | MR 1260324 | Zbl 0799.32007

[19] Lojasiewicz, S. Une propriété topologique des sous-ensembles analytiques réels, Colloques Internationaux du CNRS, Les équations aux dérivées partielles, éd. B. Malgrange (Paris 1962), Publications du CNRS, Paris Tome 117 (1963) | MR 160856 | Zbl 0234.57007

[20] Lojasiewicz, S. 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] Nemethi, A.; Zaharia, A. Milnor fibration at infinity, Indag. Math., Tome 3 (1992), pp. 323-335 | Article | MR 1186741 | Zbl 0806.57021

[22] Nowel, A.; Szafraniec, Z. On trajectories of analytic gradient vector fields, J. Differential Equations, Tome 184 (2002), pp. 215-223 | Article | MR 1929153 | Zbl 1066.58022

[23] Oleinik, O. A.; Petrovskii, I. G. On the topology of real algebraic surfaces, Amer. Math. Soc., Amer. Math. Soc. Transl., Tome 70 (1952) | MR 48095

[24] Tibăr, Mihai Regularity at infinity of real and complex polynomial functions, Singularity theory (Liverpool, 1996), Cambridge Univ. Press (London Math. Soc. Lecture Note Ser.) Tome 263 (1999), pp. xx, 249-264 | MR 1709356 | Zbl 0930.58005

[25] Van Den Dries, L.; Miller, C. Geometric categories and o-minimal structures, Duke Math. J., Tome 84 (1996), pp. 497-540 | Article | MR 1404337 | Zbl 0889.03025