Uniform existential definitions of valuations in function fields in one variable
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916962081177600 |
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| author | Becher, Karim Johannes Daans, Nicolas Dittmann, Philip |
| author_facet | Becher, Karim Johannes Daans, Nicolas Dittmann, Philip |
| contents | We study function fields of curves over a base field $K$ which is either a global field or a large field having a separable field extension of degree divisible by $4$. We show that, for any such function field, Hilbert's 10th Problem has a negative answer, the valuation rings containing $K$ are uniformly existentially definable, and finitely generated integrally closed $K$-subalgebras are definable by a universal-existential formula. In order to obtain these results, we develop further the usage of local-global principles for quadratic forms in function fields to definability of certain subrings. We include a first systematic presentation of this general method, without restriction on the characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06044 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform existential definitions of valuations in function fields in one variable Becher, Karim Johannes Daans, Nicolas Dittmann, Philip Number Theory 12L05, 12L99, 11E81, 12F20 We study function fields of curves over a base field $K$ which is either a global field or a large field having a separable field extension of degree divisible by $4$. We show that, for any such function field, Hilbert's 10th Problem has a negative answer, the valuation rings containing $K$ are uniformly existentially definable, and finitely generated integrally closed $K$-subalgebras are definable by a universal-existential formula. In order to obtain these results, we develop further the usage of local-global principles for quadratic forms in function fields to definability of certain subrings. We include a first systematic presentation of this general method, without restriction on the characteristic. |
| title | Uniform existential definitions of valuations in function fields in one variable |
| topic | Number Theory 12L05, 12L99, 11E81, 12F20 |
| url | https://arxiv.org/abs/2311.06044 |