2018/12/25 by Pantelis E. Eleftheriou, Eleftheriou, Pantelis E., Alex Savatovsky +1 · 1 citation
Mathematics · #03C64 #Advanced Topology and Set Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1812.10151
openalex publication_date 2018/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the following theorem: let \widetilde\mathcal R be an expansion of the real field \mathbb R, such that every definable set (I) is a uniform countable union of semialgebraic sets, and (II) contains a "semialgebraic chunk". Then every definable smooth function f:X⊆ \mathbb Rn→ \mathbb R with open semialgebraic domain is semialgebraic. Conditions (I) and (II) hold for various d-minimal expansions \widetilde\mathcal R = ⟨ \mathbb R, P⟩ of the real field, such as when P=2^\mathbb Z, or P⊆ \mathbb R is an iteration sequence. A generalization of the theorem to d-minimal expansions \widetilde\mathcal R of \mathbb Ran fails. On the other hand, we prove our theorem for expansions\widetilde\mathcal R of arbitrary real closed fields. Moreover, its conclusion holds for certain structures with d-minimal open core, such as ⟨ \mathbb R, \mathbb Ralg, 2^\mathbb Z⟩.