Approximation of $C^\infty $-functions without changing their zero-set

Broglia, F.; Tognoli, A.

doi:10.5802/aif.1178

Approximation of

C^{\infty}

-functions without changing their zero-set

Broglia, F. ; Tognoli, A.

Annales de l'Institut Fourier, Tome 39 (1989) no. 3, pp. 611-632.

Résumé
Abstract

On démontre que l’obstruction à approcher une fonction $C^{\infty} φ$ , dont le lieu de zéro est un ensemble algébrique ou analytique (défini par des équations globables), par des fonctions régulières ayant les mêmes zéros, est seulement la signature sur le complémentaire de $Y$ .

For a $C^{\infty}$ function $φ : M \to ℝ$ (where $M$ is a real algebraic manifold) the following problem is studied. If $φ^{- 1} (0)$ is an algebraic subvariety of $M$ , can $φ$ be approximated by rational regular functions $f$ such that $f^{- 1} (0) = φ^{- 1} (0) ?$

We find that this is possible if and only if there exists a rational regular function $g : M \to ℝ$ such that $g^{- 1} (0) = φ^{- 1} (0)$ and g(x) $\cdot φ (x) \geq 0$ for any $x$ in $ℝ^{n}$ . Similar results are obtained also in the analytic and in the Nash cases.

For non approximable functions the minimal flatness locus is also studied.

MR Zbl

DOI : 10.5802/aif.1178

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     author = {Broglia, F. and Tognoli, A.},
     title = {Approximation of $C^\infty $-functions without changing their zero-set},
     journal = {Annales de l'Institut Fourier},
     pages = {611--632},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {3},
     year = {1989},
     doi = {10.5802/aif.1178},
     mrnumber = {90k:32023},
     zbl = {0673.14017},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1178/}
}

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Broglia, F.; Tognoli, A. Approximation of $C^\infty $-functions without changing their zero-set. Annales de l'Institut Fourier, Tome 39 (1989) no. 3, pp. 611-632. doi : 10.5802/aif.1178. http://www.numdam.org/articles/10.5802/aif.1178/

Bibliographie
Cité par

[ABrT] F. Acquistapace, F. Broglia, A. Tognoli, An embedding theorem for real analytic spaces, Ann. S.N.S. Pisa, Serie IV, Vol VI, n.3 (1979), 415-426. | EuDML | Numdam | MR | Zbl

[BeT] R. Benedetti, A. Tognoli, Teoremi di approssimazione in topologia differenziale I, Boll. U.M.I., (5) 14-B (1977), 866-887. | MR | Zbl

[BiM] E. Bierstone, P.D. Milman, Arc-analytic functions, to appear. | Zbl

[BocCC-R] J. Bochnak, M. Coste, M.F. Coste-Roy, Géométrie algébrique réelle, Erg. d. Math.12, Springer, 1987. | MR | Zbl

[BorH] A. Borel, A. Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France, 89 (1961), 461-513. | EuDML | Numdam | MR | Zbl

[BrL] T. Bröcker, L. Lander, Differentiable germs and catastrophes, Cambridge Univ. Press, 1975. | Zbl

[Hiro] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., 79 (1964), 109-324. | MR | Zbl

[Hirs] M.W. Hirsh, Differential topology, Springer, 1976. | Zbl

[LT] F. Lazzeri, A. Tognoli, Alcune proprietà degli spazi algebrici, Ann. S.N.S. Pisa, 24 (1970), 597-632. | EuDML | Numdam | MR | Zbl

[M] B. Malgrange, Sur les fonctions différentiables et les ensembles analytiques, Bull. Soc. Math. France, (1963), 113-127. | EuDML | Numdam | MR | Zbl

[N1] R. Narasimhan, Introduction to the theory of analytic spaces, Lectures Notes in Math., Vol 25, Springer, 1966. | MR | Zbl

[N2] R. Narasimhan, Analysis on real and complex manifolds, Masson & Cie, Paris, 1968. | MR | Zbl

[T1] A. Tognoli, Sulla classifizione dei fibrati analitici reali, Ann. S.N.S. Pisa, 21 (4) (1967), 709-744. | EuDML | Numdam | MR | Zbl

[T2] A. Tognoli, Su una congettura di Nash, Ann. S.N.S. Pisa, 27 (4) (1973), 167-185. | EuDML | Numdam | MR | Zbl

[T3] A. Tognoli, Un teorema di approssimazione relativo, Atti Accad. Naz. Lincei Rend., (8) 40 (1973), 496-502. | Zbl

[T4] A. Tognoli, Algebraic geometry and Nash function, Institutiones Math., Vol 3, London, New York, Academic Press, 1978. | MR | Zbl

[T5] A. Tognoli, Algebraic approximation of manifolds and spaces, Sém Bourbaki, n. 548 (1979/1980). | EuDML | Numdam | Zbl

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