Abstract

Let${\cal F}$=(F; +,.,0, 1, D) be a differentially closed field. We consider the question of definability of the derivation D in reducts of${\cal F}$of the form${\cal F}$R= (F; +,.,0, 1,P)PεRwhereRis some collection of definable sets in${\cal F}$. We give examples and nonexamples and establish some criteria for definability of D. Finally, using the tools developed in the article, we prove that under the assumption of inductiveness of Th (${\cal F}$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.