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- Title
DEFINABILITY OF DERIVATIONS IN THE REDUCTS OF DIFFERENTIALLY CLOSED FIELDS.
- Authors
ASLANYAN, VAHAGN
- 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)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.
- Subjects
DIFFERENTIABLE functions; DEFINABILITY theory (Mathematical logic); MODEL theory; RECURSIVE functions; ALGORITHMIC randomness; APPLIED mathematics; ALGEBRAIC logic
- Publication
Journal of Symbolic Logic, 2017, Vol 82, Issue 4, p1252
- ISSN
0022-4812
- Publication type
Article
- DOI
10.1017/jsl.2017.54