On Nash approximation of complex analytic sets in Runge domains
type: Item
instance of: scholarly article
title: On Nash approximation of complex analytic sets in Runge domains
zbMATH Open document ID: 1304.32005
author: Janusz Adamus
author: Marcin Bilski
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1307.6851
review text: Let \(\Omega \) be an open set in \(\mathbb{C}^{n}.\) A function \(f:\Omega \rightarrow \mathbb{C}\) is a \textit{Nash function} if \(f\) locally satisfies \(P(z,f(z))=0\), where \(P(z,t)\not\equiv 0\) is a polynomial in \(z\) and \(t\). A subset \(X\subset \Omega \) is called a \textit{Nash subset} of \(\Omega \) if locally it is the zero set of a finite system of Nash functions. The main result is that every complex analytic set \(X\) in a Runge domain \(\Omega \) can be approximated (in the sense of holomorphic chains [\textit{P. Tworzewski}, Ann. Pol. Math. 62, No. 2, 177--191 (1995; Zbl 0911.32018)]) by Nash sets on any relatively compact subdomain \(\Omega _{0}\subset \Omega \). Moreover, for an arbitrary subset \(R\subset X\), the approximating sets can be chosen so that they contain \(R\cap \Omega _{0}\) if and only if there exists a Nash set \(Y\) in \( \Omega _{0}\) such that \(R\cap \Omega _{0}\subset Y\).
reviewed by: Tadeusz Krasiński
Mathematics Subject Classification ID: 32C07
zbMATH DE Number: 6362995
zbMATH Keywords: analytic set
zbMATH Keywords: Runge domain
zbMATH Keywords: approximation by Nash sets
MaRDI profile type: Publication
OpenAlex ID: W2126867252
arXiv ID: 1307.6851
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DOI: 10.1016/J.JMAA.2014.09.070
description: scientific article; zbMATH DE number 6362995