×

Approximation of \(C^{\infty}\)-functions without changing their zero-set. (English) Zbl 0673.14017

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.
Reviewer: F.Broglia

MSC:

14Pxx Real algebraic and real-analytic geometry
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] [ABrT] , , , An embedding theorem for real analytic spaces, Ann. S.N.S. Pisa, Serie IV, Vol VI, n.3 (1979), 415-426. · Zbl 0426.32001
[2] [BeT] , , Teoremi di approssimazione in topologia differenziale I, Boll. U.M.I., (5) 14-B (1977), 866-887. · Zbl 0439.58004
[3] [BiM] , , Arc-analytic functions, to appear. · Zbl 0723.32005
[4] [BocCC-R] , , , Géométrie algébrique réelle, Erg. d. Math.12, Springer, 1987. · Zbl 0633.14016
[5] [BorH] , , La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France, 89 (1961), 461-513. · Zbl 0102.38502
[6] [BrL] , , Differentiable germs and catastrophes, Cambridge Univ. Press, 1975. · Zbl 0302.58006
[7] [Hiro] , Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., 79 (1964), 109-324. · Zbl 0122.38603
[8] [Hirs] , Differential topology, Springer, 1976. · Zbl 0356.57001
[9] [LT] , , Alcune proprietà degli spazi algebrici, Ann. S.N.S. Pisa, 24 (1970), 597-632. · Zbl 0205.25201
[10] [M] , Sur les fonctions différentiables et les ensembles analytiques, Bull. Soc. Math. France, (1963), 113-127. · Zbl 0113.06302
[11] [N1] , Introduction to the theory of analytic spaces, Lectures Notes in Math., Vol 25, Springer, 1966. · Zbl 0168.06003
[12] [N2] , Analysis on real and complex manifolds, Masson & Cie, Paris, 1968. · Zbl 0188.25803
[13] [T1] , Sulla classifizione dei fibrati analitici reali, Ann. S.N.S. Pisa, 21 (4) (1967), 709-744. · Zbl 0179.28703
[14] [T2] , Su una congettura di Nash, Ann. S.N.S. Pisa, 27 (4) (1973), 167-185. · Zbl 0263.57011
[15] [T3] , Un teorema di approssimazione relativo, Atti Accad. Naz. Lincei Rend., (8) 40 (1973), 496-502. · Zbl 0299.32002
[16] [T4] , Algebraic geometry and Nash function, Institutiones Math., Vol 3, London, New York, Academic Press, 1978. · Zbl 0418.14002
[17] [T5] , Algebraic approximation of manifolds and spaces, Sém Bourbaki, n. 548 (1979/1980). · Zbl 0456.57012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.