Separately Nash and arc-Nash functions over real closed fields

摘要

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.