On Nash approximation of complex analytic sets in Runge domains (Q465397)

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 \(R\cap \Omega _{0}\subset Y\).