Existentially defining valuations in function fields over large fields
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912749330628608 |
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| author | Daans, Nicolas |
| author_facet | Daans, Nicolas |
| contents | Let $K$ be a large field such that $K[\sqrt{-1}]$ is not algebraically closed and $F/K$ a function field in one variable. Extending techniques and results from earlier work with Becher and Dittmann, we show that every valuation ring on $F$ containing $K$ is existentially definable in the language of rings with parameters from $F$. As a consequence, using a known reduction technique, we obtain the undecidability of the existential theory of $F$ in the language of rings with appropriately chosen parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04896 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existentially defining valuations in function fields over large fields Daans, Nicolas Number Theory Logic 12L99 (Primary), 11R52, 11R58, 12L05 (Secondary) Let $K$ be a large field such that $K[\sqrt{-1}]$ is not algebraically closed and $F/K$ a function field in one variable. Extending techniques and results from earlier work with Becher and Dittmann, we show that every valuation ring on $F$ containing $K$ is existentially definable in the language of rings with parameters from $F$. As a consequence, using a known reduction technique, we obtain the undecidability of the existential theory of $F$ in the language of rings with appropriately chosen parameters. |
| title | Existentially defining valuations in function fields over large fields |
| topic | Number Theory Logic 12L99 (Primary), 11R52, 11R58, 12L05 (Secondary) |
| url | https://arxiv.org/abs/2512.04896 |