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

Broglia, F., and Tognoli, A.. "Approximation of $C^\infty $-functions without changing their zero-set." Annales de l'institut Fourier 39.3 (1989): 611-632. <http://eudml.org/doc/74842>.

@article{Broglia1989,
abstract = {For a $C^\infty $ function $\phi : M\rightarrow \{\Bbb R\}$ (where $M$ is a real algebraic manifold) the following problem is studied. If $\phi ^\{-1\}(0)$ is an algebraic subvariety of $M$, can $\phi $ be approximated by rational regular functions $f$ such that $f^\{-1\}(0)=\phi ^\{-1\}(0)?$We find that this is possible if and only if there exists a rational regular function $g: M\rightarrow \{\Bbb R\}$ such that $g^\{-1\}(0)=\phi ^\{-1\}(0)$ and g(x)$\cdot \phi (x)\ge 0$ for any $x$ in $\{\Bbb R\}^n$. Similar results are obtained also in the analytic and in the Nash cases.For non approximable functions the minimal flatness locus is also studied.},
author = {Broglia, F., Tognoli, A.},
journal = {Annales de l'institut Fourier},
keywords = {approximation of -functions by regular functions; algebraic manifold; zero set},
language = {eng},
number = {3},
pages = {611-632},
publisher = {Association des Annales de l'Institut Fourier},
title = {Approximation of $C^\infty $-functions without changing their zero-set},
url = {http://eudml.org/doc/74842},
volume = {39},
year = {1989},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 3
SP - 611
EP - 632
AB - For a $C^\infty $ function $\phi : M\rightarrow {\Bbb R}$ (where $M$ is a real algebraic manifold) the following problem is studied. If $\phi ^{-1}(0)$ is an algebraic subvariety of $M$, can $\phi $ be approximated by rational regular functions $f$ such that $f^{-1}(0)=\phi ^{-1}(0)?$We find that this is possible if and only if there exists a rational regular function $g: M\rightarrow {\Bbb R}$ such that $g^{-1}(0)=\phi ^{-1}(0)$ and g(x)$\cdot \phi (x)\ge 0$ for any $x$ in ${\Bbb R}^n$. Similar results are obtained also in the analytic and in the Nash cases.For non approximable functions the minimal flatness locus is also studied.
LA - eng
KW - approximation of -functions by regular functions; algebraic manifold; zero set
UR - http://eudml.org/doc/74842
ER -