Almost analytic local algebraicity of analytic sets and functions
type: Item
instance of: scholarly article
title: Almost analytic local algebraicity of analytic sets and functions
zbMATH Open document ID: 0633.32025
DOI: 10.1007/BF01179268
author: Michael A. Buchner
author: Wojciech Kucharz
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173864
review text: Usually two set-germs \(V,W\subset ({\mathbb{R}}^ n,0)\) and two function- germs f,g: (\({\mathbb{R}}^ n,0)\to ({\mathbb{R}}^ n,0)\) are said analytically (resp. \(C^ r\), \(r=0,1,2,...)\) equivalent if there exists a local analytic (resp. \(C^ r)\) diffeomorphism \(\sigma\) : (\({\mathbb{R}}^ n,0)\to ({\mathbb{R}}^ n,0)\) such that \(\sigma (V)=W\) and \(f\circ \sigma =g\). Analytic equivalence exists only under restrictive assumptions, while \(C^ r\) equivalence looks too weak in many instances. The authors introduce a notion of ``almost analytic \(C^ r\) equivalence'' for germs of analytic sets and functions. Essentially they require \(\sigma\) to be a local \(C^ r\) diffeomorphism such that there exists a germ of analytic diffeomorphism \({\tilde \sigma}: (X,\pi^{-1}(0)) \to (X,\pi^{-1}(0))\) satisfying \(\sigma \circ \pi =\pi \circ {\tilde \sigma}\), where \(\pi\) : \(X\to {\mathbb{R}}^ n\) is an algebraic multiblow-up of \({\mathbb{R}}^ n\) at the origin. Then the authors describe a large class of analytic set-germs and function-germs, which are equivalent according to their definition to germs of algebraic sets and polynomial functions.
Mathematics Subject Classification ID: 32J99
Mathematics Subject Classification ID: 14Pxx
zbMATH DE Number: 4029960
zbMATH Keywords: affine real algebraic variety
zbMATH Keywords: real blow-up
zbMATH Keywords: almost analytic equivalence
zbMATH Keywords: germs of analytic functions
zbMATH Keywords: germs of analytic sets
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
cites work: Submanifolds and Homology of Nonsingular Real Algebraic Varieties
cites work: Algebraic approximation of structures over complete local rings
cites work: On algebraicity of global real analytic sets and functions
cites work: Q3223124
cites work: Algebraicity versus analyticity
cites work: Q3336052
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Power Series and Smooth Functions Equivalent to a Polynomial
cites work: A theorem on almost analytic equisingularity
cites work: The modified analytic trivialization of singularities
cites work: Q4189876
cites work: On classification of real singularities
cites work: Faisceaux algébriques cohérents
cites work: Equivalence of differentiable mappings and analytic mappings
description: scientific article; zbMATH DE number 4029960