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Separately Nash and arc‐Nash functions over real closed fields
Author(s) -
Kucharz Wojciech,
Kurdyka Krzysztof,
ElSiblani Ali
Publication year - 2021
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12432
Subject(s) - uncountable set , mathematics , arc (geometry) , nash equilibrium , function (biology) , field (mathematics) , combinatorics , discrete mathematics , pure mathematics , mathematical economics , geometry , countable set , evolutionary biology , biology
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.