D-minimal expansions of the real field have the zero set property

Abstract:

If $E\subseteq \mathbb R^n$ is closed and the structure $(\mathbb R,+,\cdot ,E)$ is d-minimal

(that is, in every structure elementarily equivalent to $(\mathbb R,+,\cdot ,E)$, every unary definable set is a disjoint union of open intervals and finitely many discrete sets), then for each $p\in \mathbb {N}$, there exist $C^p$ functions $f\colon \mathbb R^n\to \mathbb R$ definable in $(\mathbb R,+,\cdot ,E)$ such that $E$ is the zero set of $f$.