Strongly minimal group relics of algebraically closed valued fields
We prove Zilber's trichotomy for reducts of ACVF expanding $(K,+)$ or $(K^*, \cdot)$.
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Santiago Pinzon
Publications and source records attributed to Santiago Pinzon.
Strongly minimal reducts of ACVF
In this document we prove: Let $\mathbb K=(K,+,\cdot,v,Γ)$ be an algebraically closed valued field and let $(G,\oplus)$ be a $\mathbb K$-definable group that is either the multiplicative group or contains a finite index subgroup that is $\mathbb K$-definably isomorphic to a $\mathbb K$-definable subgroup of $(K,+)$. Then if $\mathcal G=(G,\oplus,\ldots)$ is a strongly minimal non locally modular structure definable in $\mathbb K$ and expanding $(G,\oplus)$, it interprets an infinite field. This document is the PhD thesis of the author and it was advised by professors Assaf Hasson and Alf Onshuus.
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Strongly minimal reducts of ACVF
Let $\mathbb K=(K,+,\cdot,v,Γ)$ 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.
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