We prove the following theorem: let $\tilde{\mathcal{R}}$ be an expansion of the real field $\bar{\mathbb{R}}$, 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 $\tilde{\mathcal{R}}=〈\bar{\mathbb{R}},P〉$ 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 $〈\bar{\mathbb{R}},{\mathbb{R}}_{alg},2^{\mathbb{Z}}〉$.

我们证明以下定理： $\stackrel{〜}{\mathcal{[R}}$ 扩展真实领域 $\stackrel{〜}{\mathbb{[R}}$，这样每个可定义的集合（I）是一个统一的可计数的半代数集合的并集，而（II）包含一个“半代数块”。然后每个可定义的平滑函数$F：X\subseteq {\mathbb{[R}}^{ñ}\to \mathbb{[R}$ 具有开放半代数域的是半代数的。

条件（I）和（II）适用于各种d最小展开 $\stackrel{〜}{\mathcal{[R}}=〈\stackrel{〜}{\mathbb{[R}}，P〉$ 真实的领域，例如何时 $P=2^{\mathbb{ž}}$， 要么 $P\subseteq \mathbb{[R}$是一个迭代序列。定理对d极小展开式的推广$\stackrel{〜}{\mathcal{[R}}$ 的 ${\mathbb{[R}}_{\mathrm{一种}ñ}$失败。另一方面，我们证明了扩展定理$\stackrel{〜}{\mathcal{[R}}$任意实数封闭字段。此外，其结论适用于某些具有d最小开核的结构，例如$〈\stackrel{〜}{\mathbb{[R}}，{\mathbb{[R}}_{\mathrm{一种}升G}，2^{\mathbb{ž}}〉$。