Expansions of real closed fields which introduce no new smooth functions

Abstract

We prove the following theorem: let $\tilde{\mathcal{R}}$ be an expansion of the real field $\overline{\mathbb{R}}$, 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 $f:X\subseteq {\mathbb{R}}^n\to \mathbb{R}$ with open semialgebraic domain is semialgebraic. Conditions (I) and (II) hold for various d-minimal expansions $\tilde{\mathcal{R}}=⟨\overline{\mathbb{R}},P⟩$ of the real field, such as when $P=2^{\mathbb{Z}}$, or $P\subseteq \mathbb{R}$ is an iteration sequence. A generalization of the theorem to d-minimal expansions $\tilde{\mathcal{R}}$ of ${\mathbb{R}}_{an}$ fails. On the other hand, we prove our theorem for expansions$\tilde{\mathcal{R}}$ of arbitrary real closed fields. Moreover, its conclusion holds for certain structures with d-minimal open core, such as $⟨\overline{\mathbb{R}},{\mathbb{R}}_{alg},2^{\mathbb{Z}}⟩$.

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