Coincidence of dimensions in closed ordered differential fields

Abstract

Let $\mathcal{K}=⟨\mathcal{R},\delta ⟩$ be a closed ordered differential field, in the sense of M. Singer, and $C$ its field of constants. In this note, we prove that, for sets definable in the pair $\mathcal{M}=⟨\mathcal{R},C⟩$, the $\delta$-dimension and the large dimension coincide. As an application, we characterize the definable sets in $\mathcal{K}$ that are internal to $C$ as those sets that are definable in $\mathcal{M}$ and have $\delta$-dimension $0$. We further show that, for sets definable in $\mathcal{K}$, having $\delta$-dimension $0$ does not generally imply co-analyzability in $C$ (in contrast to the case of transseries). We also point out that the coincidence of dimensions also holds in the context of differentially closed fields and in the context of transseries.