On Strongly NIP Ordered Fields

Lothar Sebastian Krapp, Salma Kuhlmann

Published 2018-10-24Version 1

The following conjecture is due to Shelah$-$Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or it admits a non-trivial definable henselian valuation, in the language $\mathcal{L}_{\mathrm{r}} = \{+,-,\cdot,0,1\}$. Inspired by this, we formulate an analogous conjecture for ordered fields in the language $\mathcal{L}_{\mathrm{or}} = \mathcal{L}_{\mathrm{r}} \cup \{<\}$. Moreover, we examine strongly NIP almost real closed fields as well as ordered Hahn fields and exhibit connections to ordered fields which are not dense in their real closure and dp-minimal ordered fields.