Extension bases in Henselian valued fields

TL;DR

This paper investigates forking behavior in Henselian valued fields, providing new conditions for parameter sets to be extension bases and demonstrating forking-dividing coincidence in ultraproducts of p-adic fields.

Contribution

It introduces broader conditions for extension bases in Henselian valued fields, including arbitrary parameter sets, extending prior results focused on maximally complete sets.

Findings

Forking coincides with dividing in ultraproducts of p-adic fields.

New sufficient conditions for parameter sets to be extension bases.

Broader applicability to arbitrary (including imaginary) bases.

Abstract

We study the behaviour of forking in valued fields, and we give several sufficient conditions for parameter sets in a Henselian valued field of residue characteristic zero to be an extension base. Notably, we consider arbitrary (potentially imaginary) bases, whereas previous related results in the literature only focus on maximally complete sets of parameters. This enables us in particular to show that forking coincides with dividing in (the imaginary expansions of) the ultraproducts of the p-adic fields.