Generic derivations on o-minimal structures
Antongiulio Fornasiero, Elliot Kaplan
TL;DR
The article develops a broad framework for generic derivations on models of a complete o-minimal theory $T$ extending $ ext{RCF}$. By introducing $T$-derivations and proving the existence of a model completion $T^ abla_G$, it obtains a distal, NIP theory with open core $T$, elimination of imaginaries, and a robust dimension theory; these results specialize to the classical CODF when $T= ext{RCF}$. The work also extends to finitely many commuting derivations, establishing the model completion $T^ abla_ ext{Delta}$ and a corresponding $ abla$-dimension theory, along with a $ abla$-cell decomposition. Collectively, the paper unifies and extends CODF-like phenomena within a general o-minimal context, linking generic derivations to dense-pair structures and open-core preservation. The results have implications for the interaction between differential algebra and o-minimality, providing tools for definable closure, imaginaries, and topology in derivative-augmented structures.
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 $δ$ 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 $T^δ_G$. The derivation in models $(\mathcal{M},δ)\models T^δ_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $\mathcal{M}$. If $T =$ RCF, then $T^δ_G$ 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 $T^δ_G$ has $T$ as its open core, that $T^δ_G$ is distal, and that $T^δ_G$ eliminates imaginaries. We also show that the theory of $T$-models with finitely many commuting $T$-derivations has a model completion.