On ordinary differentially large fields
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917877623291904 |
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| author | Sánchez, Omar León Tressl, Marcus |
| author_facet | Sánchez, Omar León Tressl, Marcus |
| contents | We provide a characterisation of differentially large fields in arbitrary characteristic and a single derivation in the spirit of Blum axioms for differentially closed fields. In the case of characteristic zero, we use these axioms to characterise differential largeness in terms of being existentially closed in the differential algebraic Laurent series ring, and we prove that any large field of infinite transcendence degree can be expanded to a differentially large field even under certain prescribed constant fields. As an application, we show that the theory of proper dense pairs of models of a complete and modelcomplete theory of large fields, is a complete theory. As a further consequence of the expansion result we show that there is no real closed and differential field that has a prime model extension in closed ordered differential fields, unless it is itself a closed ordered differential field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12977 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On ordinary differentially large fields Sánchez, Omar León Tressl, Marcus Algebraic Geometry Logic Primary: 12H05, 12E99. Secondary: 03C60 We provide a characterisation of differentially large fields in arbitrary characteristic and a single derivation in the spirit of Blum axioms for differentially closed fields. In the case of characteristic zero, we use these axioms to characterise differential largeness in terms of being existentially closed in the differential algebraic Laurent series ring, and we prove that any large field of infinite transcendence degree can be expanded to a differentially large field even under certain prescribed constant fields. As an application, we show that the theory of proper dense pairs of models of a complete and modelcomplete theory of large fields, is a complete theory. As a further consequence of the expansion result we show that there is no real closed and differential field that has a prime model extension in closed ordered differential fields, unless it is itself a closed ordered differential field. |
| title | On ordinary differentially large fields |
| topic | Algebraic Geometry Logic Primary: 12H05, 12E99. Secondary: 03C60 |
| url | https://arxiv.org/abs/2307.12977 |