Translating between NIP integral domains and topological fields
Will Johnson
TL;DR
This work establishes a tight link between NIP topological fields and NIP integral domains by showing that any definable ring topology $ au$ on an NIP field $K$ is realized as an $R$-adic topology $ au_R$ for an externally definable subring $R$ with $ ext{Frac}(R)=K$. Consequently, $ au$ is automatically a field topology and, in saturated settings, the topology can be studied via the ring $R$, which is NIP and serves as a bridge to the field’s topological structure. The authors derive that such topologies are locally bounded; in characteristic $p$ or finite dp-rank, they are gt-henselian, and for finite dp-rank, they have finite breadth (a $W_n$-topology). They also reformulate the generalized henselianity conjecture in topological terms and discuss implications of the Shelah conjecture for GHC. Altogether, the paper reduces questions about NIP rings and topological fields to questions about NIP integral domains and their $R$-adic topologies, providing a roadmap to attack henselian-type conjectures via ring-theoretic methods.
Abstract
We prove that definable ring topologies on NIP fields are closely connected to NIP integral domains. More precisely, we show that up to elementary equivalence, any NIP topological field arises from an NIP integral domain. As an application, we prove several results about definable ring topologies on NIP fields, including the following. Let $K$ be an NIP field or expansion of a field. Let $τ$ be a definable ring topology on $K$. Then $τ$ is a field topology, and $τ$ is locally bounded. If $K$ has characteristic $p$ or finite dp-rank, then $τ$ is "generalized t-henselian" in the sense of Dittman, Walsberg, and Ye, meaning that the implicit function theorem holds for polynomials. If $K$ has finite dp-rank, then $τ$ must be a topology of "finite breadth" (a $W_n$-topology). Using these techniques, we give some reformulations of the conjecture that NIP local rings are henselian.