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Nicolas Dutertre
Semi-algebraic neighborhoods of closed semi-algebraic sets
(Voisinages semi-algébriques d’ensembles semi-algébriques fermés)
Annales de l'institut Fourier, 59 no. 1 (2009), p. 429-458, doi: 10.5802/aif.2435
Article: subscription required (your ip address: 67.202.9.192) | Reviews MR 2514870 | Zbl 1174.14051
Class. Math.: 14P10, 14P25
Keywords: Tubular neighborhood, semi-algebraic sets, retraction, quasiregular approaching semi-algebraic function, quasiregular approaching semi-algebraic neighborhood
[1] V. I. Arnold, “Index of a singular point of a vector field, the Petrovski-Oleinik inequality, and mixed Hodge structures”, Funct. Anal. Appl. 12 (1978), p. 1-14 MR 498592 | Zbl 0407.57025
[2] J. Bochnak, M. Coste & M. F. Roy, Géométrie algébrique réelle, Ergebnisse der Mathematik 12, Springer-Verlag, 1987 MR 949442 | Zbl 0633.14016
[3] L. Broecker & M. Kuppe, “Integral geometry of tame sets”, Geom. Dedicata 82 (2000), p. 285-323
Article | MR 1789065 | Zbl 1023.53057
[4] S. A. Broughton, On the topology of polynomial hypersurfaces, Singularities, Part 1 (Arcata, Calif., 1981), pp.167–178, in Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1983 MR 713056 | Zbl 0526.14010
[5] M. Coste, An introduction to o-minimal geometry, in Dottorato di Recerca in Matematica, thesis, Dip. Mat. Univ. Pisa. Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000
[6] M Coste, An introduction to semi-algebraic geometry, in Dottorato di Recerca in Matematica, thesis, Dip. Mat. Univ. Pisa. Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000
[7] M. Coste & M. Reguiat,
[8] A. H. Durfee, “Neighborhoods of algebraic sets”, Trans. Amer. Math. Soc. 276 (1983), p. 517-530
Article | MR 688959 | Zbl 0529.14013
[9] N. Dutertre, “Geometrical and topological properties of real polynomial fibres”, Geom. Dedicata 105 (2004), p. 43-59
Article | MR 2057243 | Zbl 1060.14081
[10] A. Fekak, “Exposants de Lojasiewicz pour les fonctions semi-algébriques”, Ann. Polon. Math. 56 (1992), p. 123-131 MR 1159983 | Zbl 0773.14027
[11] V. M. Kharlamov, “A generalized Petrovskii inequality”, Funct. Anal. Appl. 8 (1974), p. 50-56
Article | MR 350056 | Zbl 0301.14021
[12] V. M. Kharlamov, “A generalized Petrovskii inequality II”, Funct. Anal. Appl. 9 (1975), p. 93-94 MR 399502 | Zbl 0327.14018
[13] G. M. Khimshiashvili, “On the local degree of a smooth map”, Soobshch. Akad. Nauk Gruz. SSR 85 (1977), p. 309-311 Zbl 0346.55008
[14] A. G. Khovanskii, “Index of a polynomial vector field”, Funct. Anal. Appl. 13 (1978), p. 38-45
Article | MR 527521 | Zbl 0437.57012
[15] A. G. Khovanskii, “Boundary indices of polynomial $1$-forms with homogeneous components”, St. Petersburg Math. J. 10 (1999), p. 553-575 MR 1628042 | Zbl 0990.37040
[16] K. Kurdyka, “On gradients of functions definable in o-minimal structures”, Ann. Inst. Fourier 48 (1998), p. 769-783
Cedram | MR 1644089 | Zbl 0934.32009
[17] K. Kurdyka, T. Mostowski & A. Parusinski, “Proof of the gradient conjecture of R. Thom”, Ann. of Math. (2) 152 (2000), p. 763-792
Article | MR 1815701 | Zbl 1053.37008
[18] K. Kurdyka & A. Parusinski, “$w_{\hspace{-0.55542pt}f}$-stratification of subanalytic functions and the Lojasiewicz inequality”, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), p. 129-133 MR 1260324 | Zbl 0799.32007
[19] S. Lojasiewicz,
[20] S. Lojasiewicz, “Sur les trajectoires du gradient d’une fonction analytique réelle”, Seminari di Geometria 1982–1983, Bologna (1984), p. 115-117 MR 771152 | Zbl 0606.58045
[21] A. Nemethi & A. Zaharia, “Milnor fibration at infinity”, Indag. Math. 3 (1992), p. 323-335
Article | MR 1186741 | Zbl 0806.57021
[22] A. Nowel & Z. Szafraniec, “On trajectories of analytic gradient vector fields”, J. Differential Equations 184 (2002), p. 215-223
Article | MR 1929153 | Zbl 1066.58022
[23] O. A. Oleinik & I. G. Petrovskii, On the topology of real algebraic surfaces, Amer. Math. Soc. Transl. 70, Amer. Math. Soc., 1952 MR 48095
[24] Mihai Tibăr, Regularity at infinity of real and complex polynomial functions, Singularity theory (Liverpool, 1996), Cambridge Univ. Press, 1999, p. xx, 249–264 MR 1709356 | Zbl 0930.58005
[25] L. Van den Dries & C. Miller, “Geometric categories and $o$-minimal structures”, Duke Math. J. 84 (1996), p. 497-540
Article | MR 1404337 | Zbl 0889.03025
Nicolas Dutertre
Semi-algebraic neighborhoods of closed semi-algebraic sets
(Voisinages semi-algébriques d’ensembles semi-algébriques fermés)
Annales de l'institut Fourier, 59 no. 1 (2009), p. 429-458, doi: 10.5802/aif.2435
Article: subscription required (your ip address: 67.202.9.192) | Reviews MR 2514870 | Zbl 1174.14051
Class. Math.: 14P10, 14P25
Keywords: Tubular neighborhood, semi-algebraic sets, retraction, quasiregular approaching semi-algebraic function, quasiregular approaching semi-algebraic neighborhood
Résumé - Abstract
Given a closed (not necessarly compact) semi-algebraic set $X$ in $\mathbb{R}^n$, we construct a non-negative semi-algebraic ${\mathcal{C}}^2$ function $f$ such that ${X{=}f^{-1}(0)}$ and such that for $\delta >0$ sufficiently small, the inclusion of $X$ in $f^{-1}([0,\delta ])$ is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of $X$.
Bibliography
[2] J. Bochnak, M. Coste & M. F. Roy, Géométrie algébrique réelle, Ergebnisse der Mathematik 12, Springer-Verlag, 1987 MR 949442 | Zbl 0633.14016
[3] L. Broecker & M. Kuppe, “Integral geometry of tame sets”, Geom. Dedicata 82 (2000), p. 285-323
Article | MR 1789065 | Zbl 1023.53057
[4] S. A. Broughton, On the topology of polynomial hypersurfaces, Singularities, Part 1 (Arcata, Calif., 1981), pp.167–178, in Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1983 MR 713056 | Zbl 0526.14010
[5] M. Coste, An introduction to o-minimal geometry, in Dottorato di Recerca in Matematica, thesis, Dip. Mat. Univ. Pisa. Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000
[6] M Coste, An introduction to semi-algebraic geometry, in Dottorato di Recerca in Matematica, thesis, Dip. Mat. Univ. Pisa. Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000
[7] M. Coste & M. Reguiat,
Trivialités en famille, inReal algebraic geometry (Rennes, 1991), pp.193–204, Lecture Notes in Math. 1524, Springer, Berlin, 1992 MR 1226253 | Zbl 0801.14016[8] A. H. Durfee, “Neighborhoods of algebraic sets”, Trans. Amer. Math. Soc. 276 (1983), p. 517-530
Article | MR 688959 | Zbl 0529.14013
[9] N. Dutertre, “Geometrical and topological properties of real polynomial fibres”, Geom. Dedicata 105 (2004), p. 43-59
Article | MR 2057243 | Zbl 1060.14081
[10] A. Fekak, “Exposants de Lojasiewicz pour les fonctions semi-algébriques”, Ann. Polon. Math. 56 (1992), p. 123-131 MR 1159983 | Zbl 0773.14027
[11] V. M. Kharlamov, “A generalized Petrovskii inequality”, Funct. Anal. Appl. 8 (1974), p. 50-56
Article | MR 350056 | Zbl 0301.14021
[12] V. M. Kharlamov, “A generalized Petrovskii inequality II”, Funct. Anal. Appl. 9 (1975), p. 93-94 MR 399502 | Zbl 0327.14018
[13] G. M. Khimshiashvili, “On the local degree of a smooth map”, Soobshch. Akad. Nauk Gruz. SSR 85 (1977), p. 309-311 Zbl 0346.55008
[14] A. G. Khovanskii, “Index of a polynomial vector field”, Funct. Anal. Appl. 13 (1978), p. 38-45
Article | MR 527521 | Zbl 0437.57012
[15] A. G. Khovanskii, “Boundary indices of polynomial $1$-forms with homogeneous components”, St. Petersburg Math. J. 10 (1999), p. 553-575 MR 1628042 | Zbl 0990.37040
[16] K. Kurdyka, “On gradients of functions definable in o-minimal structures”, Ann. Inst. Fourier 48 (1998), p. 769-783
Cedram | MR 1644089 | Zbl 0934.32009
[17] K. Kurdyka, T. Mostowski & A. Parusinski, “Proof of the gradient conjecture of R. Thom”, Ann. of Math. (2) 152 (2000), p. 763-792
Article | MR 1815701 | Zbl 1053.37008
[18] K. Kurdyka & A. Parusinski, “$w_{\hspace{-0.55542pt}f}$-stratification of subanalytic functions and the Lojasiewicz inequality”, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), p. 129-133 MR 1260324 | Zbl 0799.32007
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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] S. Lojasiewicz, “Sur les trajectoires du gradient d’une fonction analytique réelle”, Seminari di Geometria 1982–1983, Bologna (1984), p. 115-117 MR 771152 | Zbl 0606.58045
[21] A. Nemethi & A. Zaharia, “Milnor fibration at infinity”, Indag. Math. 3 (1992), p. 323-335
Article | MR 1186741 | Zbl 0806.57021
[22] A. Nowel & Z. Szafraniec, “On trajectories of analytic gradient vector fields”, J. Differential Equations 184 (2002), p. 215-223
Article | MR 1929153 | Zbl 1066.58022
[23] O. A. Oleinik & I. G. Petrovskii, On the topology of real algebraic surfaces, Amer. Math. Soc. Transl. 70, Amer. Math. Soc., 1952 MR 48095
[24] Mihai Tibăr, Regularity at infinity of real and complex polynomial functions, Singularity theory (Liverpool, 1996), Cambridge Univ. Press, 1999, p. xx, 249–264 MR 1709356 | Zbl 0930.58005
[25] L. Van den Dries & C. Miller, “Geometric categories and $o$-minimal structures”, Duke Math. J. 84 (1996), p. 497-540
Article | MR 1404337 | Zbl 0889.03025
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