Almost analytic local algebraicity of analytic sets and functions
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.
scientific article; zbMATH DE number 4029960
scientific article