Strongly minimal reducts of ACVF

TL;DR

This paper investigates the structure of strongly minimal reducts of algebraically closed valued fields, showing they interpret a field if they are non-locally modular and contain certain interpretable groups.

Contribution

It establishes that strongly minimal non-locally modular reducts of ACVF interpret a field, extending understanding of their algebraic structure.

Findings

Strongly minimal non-locally modular reducts interpret a field.

Interpretable groups locally isomorphic to (K,+) or (K,·) imply field interpretation.

Provides a strategy to prove similar results without definable group operations.

Abstract

Let $\mathbb K=(K,+,\cdot,v,\Gamma)$ be a valued algebraically closed field of characteristic and $(G,\oplus)$ be a $\mathcal K$-interpretable group that is either locally isomorphic to $(K,+)$ or to $(K,\cdot)$. Then if $\mathcal G=(G,\oplus,\ldots)$ is a strongly minimal non locally modular structure intepretable in $\mathbb K$, it interprets a field. We also present an strategy for proving the same without the assumption of having a definable group operation.