# Generic derivations on o-minimal structures

## TL;DR

This paper introduces a general framework for derivations compatible with o-minimal structures, extending known theories like CODF, and establishes their model-theoretic properties such as distality and elimination of imaginaries.

## Contribution

It defines the notion of a T-derivation in o-minimal theories, constructs a model completion T^δ_G, and generalizes properties of closed ordered differential fields to a broader setting.

## Key findings

- The theory T^δ_G has a model completion with a generic derivation.
- T^δ_G is distal and eliminates imaginaries.
- When T=RCF, T^δ_G corresponds to CODF.

## Abstract

Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $\delta$ on models $\mathcal{M}\models T$. We introduce the notion of a $T$-derivation: a derivation which is compatible with the $L(\emptyset)$-definable $\mathcal{C}^1$-functions on $\mathcal{M}$. We show that the theory of $T$-models with a $T$-derivation has a model completion $T^\delta_G$. The derivation in models $(\mathcal{M},\delta)\models T^\delta_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $\mathcal{M}$. If $T =$ RCF, then $T^\delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^\delta_G$ has $T$ as its open core, that $T^\delta_G$ is distal, and that $T^\delta_G$ eliminates imaginaries. We also show that the theory of $T$-models with finitely many commuting $T$-derivations has a model completion.