51. Monotone $T$-convex $T$-differential fields
- Author
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Kaplan, Elliot and Pynn-Coates, Nigel
- Subjects
Mathematics - Logic - Abstract
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., Comment: 27 pages; revised according to suggestions by the referee, including equivalent simpler definition of $T^{\partial}$-henselianity, comparing more thoroughly $T^{\partial}$-henselianity and differential-henselianity, and small corrections; added Theorem 4.15 and Corollary 6.8
- Published
- 2023