Approximation in compact Nash manifolds. (English) Zbl 0873.32007

Coste, Michel; Ruiz, Jesús M.; Shiota, Masahiro

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)

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