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DEFINABILITY OF DERIVATIONS IN THE REDUCTS OF DIFFERENTIALLY CLOSED FIELDS
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- Journal:
- The Journal of Symbolic Logic / Volume 82 / Issue 4 / December 2017
- Published online by Cambridge University Press:
- 09 January 2018, pp. 1252-1277
- Print publication:
- December 2017
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- Article
- Export citation
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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)P ε R where R is 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.
