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Approximation in compact Nash manifolds. (English) Zbl 0873.32007
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.

##### 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
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