Definability of derivations in the reducts of differentially closed fields

TL;DR

This paper investigates when the derivation in a differentially closed field can be defined within reducts formed by adding definable sets, providing criteria and model-theoretic conditions for such definability.

Contribution

It establishes criteria for the definability of the derivation in reducts of differentially closed fields and links model completeness to definability under inductiveness assumptions.

Findings

Model completeness is necessary for derivation definability under inductiveness.

Criteria for definability of the derivation in reducts are provided.

Examples and non-examples illustrate the conditions for definability.

Abstract

Let $\mathcal{F}=(F;+,\cdot,0,1,D)$ be a differentially closed field. We consider the question of definability of the derivation $D$ in reducts of $\mathcal{F}$ of the form $\mathcal{F}_{R}=(F;+,\cdot,0,1,P)_{P \in R}$ where $R$ is a collection of definable sets in $\mathcal{F}$. We give examples and non-examples and establish some criteria for definability of $D$. Finally, using the tools developed in the paper we prove that under the assumption of inductiveness of $Th(\mathcal{F}_{R})$ model completeness is a necessary condition for definability of $D$. This can be seen as part of a broader project where one is interested in finding Ax-Schanuel type inequalities (or predimension inequalities) for differential equations.