Definability of derivations in the reducts of differentially closed fields

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.

34 pages. New results and clarifications in some proofs added

References in corpus (1)

• The theory of the exponential differential equations of semiabelian varieties

Cited by in corpus (2)

• Ax-Schanuel and strong minimality for the $j$-function
• Ax-Schanuel for Linear Differential Equations