DEFINABILITY OF DERIVATIONS IN THE REDUCTS OF DIFFERENTIALLY CLOSED FIELDS

Let F = (F; +, ·, 0, 1, D) be a differentially closed field. We consider the question of definability of the derivation D in reducts of F of the form F R = (F; +, ·, 0, 1, P) P∈R where R is some collection of definable sets in 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(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.