2020/02/28 by Eleftheriou, Pantelis E., Sanchez, Omar Leon, Regnault, Nathalie · 1 citation
#03C60 #03C98 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2002.12929
Let \mathcal K=⟨\mathcal R, δ⟩ 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 δ-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 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.