Semi-algebraic neighborhoods of closed semi-algebraic sets

Dutertre, Nicolas

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 429-458.

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

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

[1] Arnold, V. I. 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 | Zbl

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

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

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

[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, Lecture Notes in Math., 1524, Springer, Berlin, 1992 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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), 117, Publications du CNRS, Paris, 1963 | MR | Zbl

[20] Lojasiewicz, S. Sur les trajectoires du gradient d’une fonction analytique réelle, Seminari di Geometria 1982–1983, Bologna (1984), pp. 115-117 | MR | Zbl

[21] Nemethi, A.; Zaharia, A. Milnor fibration at infinity, Indag. Math., Volume 3 (1992), pp. 323-335 | DOI | MR | Zbl

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

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

[24] Tibăr, Mihai 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 | Zbl

[25] Van den Dries, L.; Miller, C. Geometric categories and $o$-minimal structures, Duke Math. J., Volume 84 (1996), pp. 497-540 | DOI | MR | Zbl