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arXiv · 2410.22442

Strongly Minimal Relics of T-convex Fields

Benjamin Castle·Assaf Hasson
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Abstract

Generalizing previous work on algebraically closed valued fields (ACVF) and o-minimal fields, we study strongly minimal relics of real closed valued fields (RCVF), and more generally T-convex expansions of o-minimal fields. Our main result (replicating the o-minimal setting) is that non-locally modular strongly minimal definable relics of T-convex fields must be two-dimensional. We also continue our work on reducing the trichotomy for general relics of a structure to just the relics of certain distinguished sorts. To this end, we prove that the trichotomy for definable RCVF-relics implies the trichotomy for interpretable RCVF-relics, and also that the trichotomy for relics of o-minimal fields implies the trichotomy for relics of any dense o-minimal structure. Finally, we introduce the class of differentiable Hausdorff geometric fields (containing o-minimal fields and various valued fields), and give a general treatment of the trichotomy for one-dimensional relics of such fields (namely, reducing the trichotomy for one-dimensional relics to an axiomatic condition on the field itself).

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Benjamin Castle, Assaf Hasson. 2024-10-29. Strongly Minimal Relics of T-convex Fields. https://arxiv.org/abs/2410.22442

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