Universally defining subrings in function fields
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911633636327424 |
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| author | Daans, Nicolas Dittmann, Philip |
| author_facet | Daans, Nicolas Dittmann, Philip |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02749 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universally defining subrings in function fields Daans, Nicolas Dittmann, Philip Number Theory Logic 12L99 (primary), 11E81, 12L05, 12F20 (secondary) 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. |
| title | Universally defining subrings in function fields |
| topic | Number Theory Logic 12L99 (primary), 11E81, 12L05, 12F20 (secondary) |
| url | https://arxiv.org/abs/2404.02749 |