[Paper Review] Dp-finite fields IV: the rank 2 picture

Will Johnson|arXiv (Cornell University)|Mar 20, 2020

Advanced Topology and Set Theory5 references4 citations

TL;DR

This paper investigates dp-finite fields of characteristic 0 and dp-rank 2, proving the existence and uniqueness of a definable V-topology under instability, despite a counterexample showing the canonical topology need not be a V-topology. The authors introduce DV-topologies—defined by a valuation and derivation—and characterize the canonical topology in non-V-topology cases, advancing the classification of dp-finite fields and the Shelah conjecture for finite dp-rank.

ABSTRACT

We investigate fields of characteristic 0 and dp-rank 2. While we do not obtain a classification, we prove that any unstable field of characteristic 0 and dp-rank 2 admits a unique definable V-topology. If this statement could be generalized to higher ranks, we would obtain the expected classification of fields of finite dp-rank. We obtain the unique definable V-topology by investigating the "canonical topology" defined in earlier work. Contrary to earlier expectations, the canonical topology need not be a V-topology. However, we are able to characterize the canonical topology (on fields of dp-rank 2 and characteristic 0) in terms of differential valued fields. This differential valued structure is obtained through a partial classification of "2-inflators," a sort of generalized valuation that arises naturally in fields of finite rank. Additionally, we give an example of a dp-rank 2 expansion of ACVF with a definable set of full rank and empty interior. This example interferes with certain strategies for proving the henselianity conjecture.

Motivation & Objective

• To understand the structure of unstable dp-finite fields of dp-rank 2 and characteristic 0.
• To investigate whether the canonical topology on such fields is a V-topology, as conjectured in prior work.
• To characterize the canonical topology when it fails to be a V-topology, using a combination of valuation theory and derivations.
• To provide a counterexample showing the canonical topology need not be a V-topology, challenging earlier expectations.
• To lay groundwork for generalizing the classification of dp-finite fields to higher ranks by analyzing 2-inflators and their algebraic structure.

Proposed method

• Analyzes the canonical topology on a monster model of an unstable dp-rank 2 field via definable sets of full rank.
• Introduces the concept of a DV-topology, defined by sets of the form $ B_{a,b, u} = \{x : \operatorname{val}(x-a) > \nu \text{ and } \operatorname{val}(\delta x - b) > \nu\} $, where $\delta$ is a derivation.
• Uses the $K$-infinitesimals $J_K$ to study the topology and relate them to multi-valuation rings.
• Applies model-theoretic techniques to 2-inflators—generalized valuations arising in finite-rank structures—proving their partial classification.
• Constructs a counterexample in $\mathrm{ACVF}_{0,0}$ with a full-rank definable set of empty interior, showing the canonical topology is not a V-topology.
• Proves that in odd characteristic, the canonical topology on dp-rank 2 fields is always a V-topology, establishing a positive case for the valuation conjecture.

Experimental results

Research questions

• RQ1 Does every unstable dp-finite field of dp-rank 2 and characteristic 0 admit a unique definable V-topology?
• RQ2 Can the canonical topology on such fields be fully characterized when it is not a V-topology?
• RQ3 What is the role of derivations and valuations in defining alternative topologies (DV-topologies) in the absence of a V-topology?
• RQ4 Does the existence of a definable set of full rank and empty interior in a dp-rank 2 structure interfere with the henselianity conjecture?
• RQ5 To what extent can the classification of dp-finite fields be extended beyond rank 1 by generalizing the results on canonical topologies and inflators?

Key findings

• For any unstable field $K$ of characteristic 0 and dp-rank 2, the canonical topology is definable and there exists a unique definable non-trivial V-topology.
• The canonical topology is not necessarily a V-topology; a counterexample is constructed in $\mathrm{ACVF}_{0,0}$ with a definable set of full rank but empty interior.
• When the canonical topology is not a V-topology, it is precisely a DV-topology defined by a valuation $\operatorname{val}: K \to \Gamma$ and a derivation $\delta: \mathbb{K} \to \mathbb{K}$, with basis sets $B_{a,b,\gamma}$.
• The non-emptiness of all $B_{a,b,\gamma}$ expresses a form of independence between the derivation and the valuation, ensuring the topology is well-behaved.
• In odd characteristic, the canonical topology on dp-rank 2 fields is always a V-topology, confirming the valuation conjecture in this case.
• The canonical topology is governed by a 2-inflator, and its structure is fully characterized via a model-theoretic analysis of generalized valuations and their interaction with derivations.