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