Approximation of <span class="mathjax-formula">$C^\infty $</span>-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 90k:32023 | Zbl 0673.14017

DOI : https://doi.org/10.5802/aif.1178

BibTeX
RIS

@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},
     zbl = {0673.14017},
     mrnumber = {90k:32023},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1178/}
}

TY  - JOUR
AU  - Broglia, F.
AU  - Tognoli, A.
TI  - Approximation of $C^\infty $-functions without changing their zero-set
JO  - Annales de l'Institut Fourier
PY  - 1989
DA  - 1989///
SP  - 611
EP  - 632
VL  - 39
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1178/
UR  - https://zbmath.org/?q=an%3A0673.14017
UR  - https://www.ams.org/mathscinet-getitem?mr=90k:32023
UR  - https://doi.org/10.5802/aif.1178
DO  - 10.5802/aif.1178
LA  - en
ID  - AIF_1989__39_3_611_0
ER  -

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/

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