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Separately Nash and arc‐Nash functions over real closed fields
Bulletin of the London Mathematical Society ( IF 0.8 ) Pub Date : 2020-11-09 , DOI: 10.1112/blms.12432
Wojciech Kucharz 1 , Krzysztof Kurdyka 2 , Ali El‐Siblani 3
Affiliation  

Let R be a real closed field. We prove that if R is uncountable, then any separately Nash (respectively, arc‐Nash) function defined over R is semialgebraic (respectively, continuous semialgebraic). To complete the picture, we provide an example showing that the assumption on R to be uncountable cannot be dropped. Moreover, even if R is uncountable but non‐Archimedean, then the shape of the domain of a separately Nash function matters for the conclusion. For R = R , we prove that arc‐Nash functions coincide with arc‐analytic semialgebraic functions.

中文翻译:

在实数封闭字段上分别具有Nash和arc-Nash函数

[R 成为一个真正的封闭领域。我们证明 [R 是不可数的,则在其上定义的任何单独的Nash(分别是arc-Nash)函数 [R 是半代数的(分别是连续半代数的)。为了使图片更完整,我们提供了一个示例,该示例表明 [R 不可数不能被丢弃。而且,即使 [R 是不可数的但不是阿基米德的,则单独的Nash函数的域的形状对于结论很重要。为了 [R = [R ,我们证明arc-Nash函数与arc-analytic半代数函数一致。
更新日期:2020-11-09
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