C-minimal fields have the exchange property

Will Johnson

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arXiv · 2403.17478

C-minimal fields have the exchange property

Will Johnson

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Abstract

We show that C-minimal fields (i.e., C-minimal expansions of ACVF) have the exchange property, answering a question of Haskell and Macpherson. Additionally, we strengthen some theorems of Cubides Kovacsics and Delon on C-minimal fields. First, we show that definably complete C-minimal fields of characteristic 0 have generic differentiability. Second, we show that if the induced structure on the residue field is a pure ACF, then polynomial boundedness holds. In fact, polynomial boundedness can only fail if there are unexpected definable automorphisms of the multiplicative group of the residue field.

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Will Johnson. 2024-03-26. C-minimal fields have the exchange property. https://arxiv.org/abs/2403.17478

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