Arithmetic definability of henselian valuations transfers across the tilting correspondence, and perfectoid cases are parameter-free when definable.
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Ordered henselian valued fields have relative field quantifier elimination in the Denef-Pas language with orders and their definable sets are Borel in the order topology.
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Definability via the tilting correspondence
Arithmetic definability of henselian valuations transfers across the tilting correspondence, and perfectoid cases are parameter-free when definable.
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Ordered henselian valued fields: definability and Borel sets
Ordered henselian valued fields have relative field quantifier elimination in the Denef-Pas language with orders and their definable sets are Borel in the order topology.