A real algebraic variety W of dimension m is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of R^m. Let l be any nonnegative integer. We prove that every map of class C^l from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the C^l topology by piecewise-regular maps of class C^k, where k is an arbitrary integer greater than or equal to l. Next we derive consequences regarding algebraization of topological vector bundles.

中文翻译：

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通过分段正则映射逼近

如果一个维数为 m 的实代数簇 W 的每个点都有一个 Zariski 开邻域，该邻域与 R^m 的一个 Zariski 开子集双正则同构，则称其是一致有理的。令 l 为任何非负整数。我们证明了从实代数簇的紧凑子集到一致有理实代数簇的每个类 C^l 的映射都可以通过类 C^k 的分段正则映射在 C^l 拓扑中近似，其中 k 是大于或等于 l 的任意整数。接下来我们推导出关于拓扑向量丛代数化的结果。