On some NIP Fragments of Fields

TL;DR

This paper investigates NIP formulas in fields and valued fields, providing new proofs of NIP properties in separably closed valued fields, and exploring transfer principles of NIP formulas between valued fields, residue fields, and value groups.

Contribution

It offers new proofs of NIP in separably closed valued fields and establishes transfer results for NIP formulas in henselian valued fields under certain conditions.

Findings

Separable closed valued fields of any characteristic are NIP.

Henselian valued fields with NIP existential formulas are henselian.

Transfer of NIP formulas depends on residue field and value group properties.

Abstract

In this note we study sets of NIP formulas in some theories of fields and valued fields, with a special focus on the sets of quantifier-free and existential formulas. First, we give a new proof of the fact that Separably Closed Valued Fields of any characteristic and any imperfection degree are NIP, and use this result to fill some gaps of a proof of the so-called NIP Transfer Theorem for henselian valued fields of equal characteristic. Second, we prove a variant of a theorem of Johnson: every positive characteristic valued field whose existential formulas are NIP is henselian. Finally, we set the ground for the finer question of transfer of NIP formulas of valued fields with bounded quantifier rank. Namely, we prove that for any henselian equicharacteristic valued field, any formula of quantifier rank at most $n\geq 1$ is NIP if and only if the same is true for the residue field and the value group, provided that the valued field is separably defectless Kaplansky and conditional on a multi-variable generalization of a well known statement about indiscernible sequences of singletons in ac-valued fields.