Monotone $T$-convex $T$-differential fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913678177075200 |
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| author | Kaplan, Elliot Pynn-Coates, Nigel |
| author_facet | Kaplan, Elliot Pynn-Coates, Nigel |
| contents | Let $T$ be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that $T$ is power bounded. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring $\mathcal{O}$ and a $T$-derivation $\partial$ such that $\partial$ is monotone, i.e., weakly contractive with respect to the valuation induced by $\mathcal{O}$. We show that the theory of monotone $T$-convex $T$-differential fields, i.e., the common theory of such $K$, has a model completion, which is complete and distal. Among the axioms of this model completion, we isolate an analogue of henselianity that we call $T^{\partial}$-henselianity. We establish an Ax--Kochen/Ershov theorem and further results for monotone $T$-convex $T$-differential fields that are $T^{\partial}$-henselian. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_13951 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Monotone $T$-convex $T$-differential fields Kaplan, Elliot Pynn-Coates, Nigel Logic Let $T$ be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that $T$ is power bounded. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring $\mathcal{O}$ and a $T$-derivation $\partial$ such that $\partial$ is monotone, i.e., weakly contractive with respect to the valuation induced by $\mathcal{O}$. We show that the theory of monotone $T$-convex $T$-differential fields, i.e., the common theory of such $K$, has a model completion, which is complete and distal. Among the axioms of this model completion, we isolate an analogue of henselianity that we call $T^{\partial}$-henselianity. We establish an Ax--Kochen/Ershov theorem and further results for monotone $T$-convex $T$-differential fields that are $T^{\partial}$-henselian. |
| title | Monotone $T$-convex $T$-differential fields |
| topic | Logic |
| url | https://arxiv.org/abs/2309.13951 |