Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\ldots,A_{M}\in\mathscr{R}$. The \emph{$C^{m}$-closure} of $(A_{1},\ldots,A_{M})$, denoted here by $[A_{1},\ldots,A_{M};C^{m}]$, is the ideal of all $f\in \mathscr{R}$ expressible in the form $f=F_{1}A_{1}+\cdots +F_{M}A_{M}$ with each $F_{i}\in C^{m}(\mathbb{R}^{n})$. In this paper we exhibit an algorithm for computing generators for $[A_{1},\ldots,A_{M};C^{m}]$.

中文翻译：

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理想的$ C ^ m $闭包的生成器

令$ \ mathscr {R} $表示$ \ mathbb {R} ^ {n} $上的实多项式环。固定$ m \ geq 0 $，然后让$ A_ {1}，\ ldots，A_ {M} \ in \ mathscr {R} $。$（A_ {1}，\ ldots，A_ {M}）$的\ emph {$ C ^ {m} $-closure}，在这里用$ [A_ {1}，\ ldots，A_ {M}表示； C ^ {m}] $是\ mathscr {R} $中所有$ f \的理想形式，其形式为$ f = F_ {1} A_ {1} + \ cdots + F_ {M} A_ {M} C $ {m}（\ mathbb {R} ^ {n}）$中的每个$ F_ {i} \中的$。在本文中，我们展示了一种用于为$ [A_ {1}，\ ldots，A_ {M}; C ^ {m}] $计算生成器的算法。