Coste, Michel; Ruiz, Jesús M.; Shiota, Masahiro Approximation in compact Nash manifolds. (English) Zbl 0873.32007 Am. J. Math. 117, No. 4, 905-927 (1995). Let \(\Omega\subset \mathbb R^n\) be a compact Nash manifold, \(\mathcal N(\Omega)\) and \(\mathcal O(\Omega)\) the rings of global Nash and global analytic functions on \(\Omega.\) The main result of this paper is the Approximation Theorem: Let \(F_1,\dots,F_q:\Omega\times\mathbb R^p\to \mathbb R\) be Nash functions, and \(f_1,\dots,f_p\in \mathcal O(\Omega)\) global analytic functions such that \(y=(f_1(x),\dots,f_p(x))\) is a solution of the system \(F_1(x,y)=\dots=F_q(x,y)=0.\) Then there are Nash functions \(g_1,\dots,g_p\in\mathcal N(\Omega),\) arbitrarily close to \(f_1,\dots,f_p\) in the Whitney topology, such that \(y=(g_1(x),\dots,g_p(x))\) is also a solution of that system. The proof is based on the so-called Néron desingularization. Using the Approximation Theorem, one can solve in the affirmative several problems on global Nash functions that have been open for many years. Reviewer: V.A.Chernecky (Odessa) Cited in 7 ReviewsCited in 11 Documents MSC: 32C07 Real-analytic sets, complex Nash functions 58A07 Real-analytic and Nash manifolds 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14P20 Nash functions and manifolds Keywords:Nash manifold; global analytic functions; Nash functions; Néron desingularization; approximation theorem PDF BibTeX XML Cite \textit{M. Coste} et al., Am. J. Math. 117, No. 4, 905--927 (1995; Zbl 0873.32007) Full Text: DOI