Ordered henselian valued fields: definability and Borel sets
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913020732506112 |
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| author | Krapp, Lothar Sebastian Vermeulen, Floris |
| author_facet | Krapp, Lothar Sebastian Vermeulen, Floris |
| contents | We firstly show that due to their resplendency ordered henselian valued fields admit relative field quantifier elimination in the Denef--Pas language expanded by linear orders in the field and residue field sort. Secondly, we deduce from a dimensionality reduction theorem that any set definable over an ordered henselian valued field is a Borel set with respect to the order topology. Our results are contextualised within Shelah's classification conjecture of NIP fields and its connections to the study of definable henselian valuations and the Fundamental Theorem of Statistical Learning. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_08638 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ordered henselian valued fields: definability and Borel sets Krapp, Lothar Sebastian Vermeulen, Floris Logic Commutative Algebra Primary 03C64, 03C10, Secondary 12J10, 12L12, 12J25, 12J15 We firstly show that due to their resplendency ordered henselian valued fields admit relative field quantifier elimination in the Denef--Pas language expanded by linear orders in the field and residue field sort. Secondly, we deduce from a dimensionality reduction theorem that any set definable over an ordered henselian valued field is a Borel set with respect to the order topology. Our results are contextualised within Shelah's classification conjecture of NIP fields and its connections to the study of definable henselian valuations and the Fundamental Theorem of Statistical Learning. |
| title | Ordered henselian valued fields: definability and Borel sets |
| topic | Logic Commutative Algebra Primary 03C64, 03C10, Secondary 12J10, 12L12, 12J25, 12J15 |
| url | https://arxiv.org/abs/2604.08638 |