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

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Abstract (VO)
Résumé

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.

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 $Y$ .

MR Zbl

DOI : 10.5802/aif.1178

@article{AIF_1989__39_3_611_0,
     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://geodesic.mathdoc.fr/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

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