On Coincidence of Dimensions in Closed Ordered Differential Fields

Abstract

Let $\mathcal{K}=⟨\mathcal{R},\mathit{\delta }⟩$ be a closed ordered differential field, in the sense of Singer, and let C be its field of constants. In this note, we prove that, for sets definable in the pair $\mathcal{M}=⟨\mathcal{R},C⟩$, the δ-dimension of Brihaye, Michaux, and Rivière and the large dimension from of Eleftheriou, Günaydin, and Hieronymi 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 δ-dimension 0. We further show that, for sets definable in $\mathcal{K}$, having δ-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.