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Preprint Number 2626
2626. Benjamin Castle, Assaf Hasson, Jinhe Ye Zilber's Trichotomy in Hausdorff Geometric Structures E-mail: Submission date: 3 May 2024 Abstract:
We give a new axiomatic treatment of the Zilber trichotomy, and use it to
complete the proof of the trichotomy for relics of algebraically closed fields,
i.e., reducts of the ACF-induced structure on ACF-definable sets. More
precisely, we introduce a class of geometric structures equipped with a
Hausdorff topology, called Hausdorff geometric structures. Natural
examples include the complex field; algebraically closed valued fields;
o-minimal expansions of real closed fields; and characteristic zero Henselian
fields (in particular p-adically closed fields). We then study the Zilber
trichotomy for relics of Hausdorff geometric structures, showing that under
additional assumptions, every non-locally modular strongly minimal relic on a
real sort interprets a one-dimensional group. Combined with recent results,
this allows us to prove the trichotomy for strongly minimal relics on the real
sorts of algebraically closed valued fields. Mathematics Subject Classification: 0C345, 14A99 Keywords and phrases: |
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