Monotone $T$-convex $T$-differential fields

07.06.2023 15:00 - 16:30

N. Pynn-Coates (Ohio State U, US)

I will discuss joint work with Elliot Kaplan on the model theory of ordered differential Hahn fields and similar ordered valued differential fields that are expanded by additional o-minimal structure, such as analytic structure.

More precisely, let \(T\) be a power bounded o‑minimal theory extending the theory of real closed fields. A \(T\)-convex \(T\)-differential field is an expansion of a model of \(T\) by a valuation and a derivation, each of which is compatible with the o-minimal structure, the former in the \(T\)-convex sense of van den Dries–Lewenberg and the latter in the sense of Fornasiero–Kaplan. When \(T=T_{\textrm{an}}\), the theory of the real field with restricted analytic functions, we can expand an ordered differential Hahn field to a \(T_{\textrm{an}}\)‑convex \(T_{\textrm{an}}\)‑differential field, in which case the derivation is monotone, i.e., weakly contractive with respect to the valuation (monotone differential Hahn fields were studied earlier by Scanlon and Hakobyan). We show that any other monotone \(T_{\textrm{an}}\)-convex \(T_{\textrm{an}}\)-differential field is elementarily equivalent to such an ordered differential Hahn field, which follows from a more general Ax–Kochen/Ershov type theorem for monotone \(T\)-convex \(T\)-differential fields. A key step is isolating an appropriate analogue of henselianity in this setting.




SR 10, 1. Stock, Koling. 14-16, 1090 Wien