Generic derivations, differential largeness, and NTP$_2$
Elliot Kaplan, Christoph Kesting
TL;DR
The paper investigates the relationship between generic derivations on algebraically bounded structures and differential largeness, showing that for a single derivation genericity and differential largeness coincide on éz-fields and that NTP$_2$ is preserved when expanding by a generic derivation. It develops a topological axiomatization based on the étale open topology to connect these notions and establishes transfer theorems: if the base theory has $NTP_2$ or NATP, the derivative-expanded theory $T^\delta_g$ retains the same property. The results unify two strands of tameness theory in differential algebra, extend to multiple commuting derivations, and provide tools for analyzing differential expansions of algebraically bounded structures. Overall, the work clarifies when differential expansions preserve model-theoretic tameness and offers a framework linking genericity with differential largeness via étale-topological methods.
Abstract
We compare Fornasiero and Terzo's framework of generic derivations on algebraically bounded structures with León Sánchez and Tressl's differentially large fields. We show in the case of a single derivation that genericity and differential largeness coincide for éz-fields, as introduced by Walsberg and Ye. We also show that an NTP$_2$ algebraically bounded structure remains NTP$_2$ after expanding by a generic derivation.