Expansions of real closed fields which introduce no new smooth functions

Abstract

We prove the following theorem: let $\widetilde{\mathcal R}$ 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 $\widetilde{\mathcal R} = \langle \overline{\mathbb R}, P\rangle$ 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 $\widetilde{\mathcal R}$ of $\mathbb R_{an}$ 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 $\langle \overline{\mathbb R}, \mathbb R_{alg}, 2^\mathbb Z\rangle$.