Approximation of \(C^{\infty}\)-functions without changing their zero-set
type: Item
instance of: scholarly article
title: Approximation of \(C^{\infty}\)-functions without changing their zero-set
zbMATH Open document ID: 0673.14017
author: Alberto Tognoli
author: Fabrizio Broglia
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74842
review text: For a \(C^{\infty}\) function \(\phi:\quad M\to {\mathbb{R}}\) (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 \(g:\quad M\to {\mathbb{R}}\) such that \(g^{-1}(0)=\phi^{- 1}(0)\) and g(x)\(\cdot \phi (x)\geq 0\) for any x in \({\mathbb{R}}^ n\). Similar results are obtained also in the analytic and in the Nash cases. For non approximable functions the minimal flatness locus is also studied.
Mathematics Subject Classification ID: 14Pxx
zbMATH DE Number: 4101360
zbMATH Keywords: approximation of \(C^{\infty }\)-functions by regular functions
zbMATH Keywords: algebraic manifold
zbMATH Keywords: zero set
reviewed by: Fabrizio Broglia
MaRDI profile type: Publication
OpenAlex ID: W2315118412
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DOI: 10.5802/AIF.1178
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 4101360