ARTIN–SCHREIER EXTENSIONS AND COMBINATORIAL COMPLEXITY IN HENSELIAN VALUED FIELDS

Abstract We give explicit formulas witnessing IP, IP $_{\!n}$ , or TP2 in fields with Artin–Schreier extensions. We use them to control p -extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP $_{\!n}$ context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP $_{\!n}$ henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.