Artin approximation and other related approximation results are used in various areas. The traditional formulation of such results is restricted to filtrations by powers of ideals, $\{I^j\}$, and to Noetherian rings. In this paper we extend several approximation results both to rather general filtrations and to $C^r$-rings, for $2\le r\le\infty$. As an auxiliary step we establish the surjectivity of the completion map for rings of $C^\infty$ functions, for a very broad class of filtrations.

中文翻译：

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一般过滤和 C 方程的 Artin-Tougeron 型近似结果

Artin 近似和其他相关的近似结果用于各个领域。此类结果的传统公式仅限于通过理想的幂过滤 $\{I^j\}$ 和诺特环。在本文中，我们将几个近似结果扩展到相当一般的过滤和 $C^r$-rings，对于 $2\le r\le\infty$。作为辅助步骤，我们为非常广泛的过滤类别建立了 $C^\infty$ 函数环的完成映射的满射性。