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arXiv · 2404.02749

Universally defining subrings in function fields

Nicolas Daans·Philip Dittmann
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Abstract

We establish that all rings of $S$-integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local fields. That is, we show that the complement of such a ring of $S$-integers is always a diophantine set. As a technical tool, we use a reciprocity exact sequence for quadratic Witt groups in function fields over almost arbitrary base fields (of any characteristic), which is new and of potentially independent interest.

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Nicolas Daans, Philip Dittmann. 2024-04-03. Universally defining subrings in function fields. https://doi.org/10.1515/crelle-2026-0026

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