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.

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