Article Dans Une Revue International Mathematics Research Notices Année : 2023

Regulous Functions over Real Closed Fields

Résumé

Abstract Let $X$ be a quasi-projective algebraic variety over a real closed field $R$, and let $f \colon U \to R$ be a function defined on an open subset $U$ of the set $X(R)$ of $R$-rational points of $X$. Assume that either the function $f$ is locally semialgebraic or the field $R$ is uncountable. If for every irreducible algebraic curve $C \subset X$ the restriction $f|_{U \cap C}$ is continuous and admits a rational representation, then $f$ is continuous and admits a rational representation. There are also suitable versions of this theorem with algebraic curves replaced by algebraic arcs. Heretofore, results of such a type have been known only for $R={\mathbb{R}}$. The transition from ${\mathbb{R}}$ to $R$ is not automatic at all and requires new methods.

Dates et versions

hal-04733637 , version 1 (12-10-2024)

Identifiants

Citer

Wojciech Kucharz, Krzysztof Kurdyka. Regulous Functions over Real Closed Fields. International Mathematics Research Notices, 2023, 2023 (3), pp.2649-2664. ⟨10.1093/imrn/rnab326⟩. ⟨hal-04733637⟩
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