Imaginaries in valued differential fields II: Computing canonical bases
In 2000, Scanlon described a theory of existentially closed differential fields where the derivation is contractive: v(d(x)) ≥ v(x), for all x. He also proved a quantifier elimination result for this theory. Around the same time, Haskell, Hrushovski and Macpherson classified all the quotients of definable sets by definable equivalence relations in a algebraically closed valued field by proving elimination of imaginaries (relative to certain quotients of the linear group). In analogy with the pure field situation where elimination of imaginaries for differentially closed fields can be derived from elimination of imaginaries in the underlying algebraically closed field, it was conjectured that Scanlon’s theory of existentially closed contractive valued differential fields also eliminated imaginaries relatively to those same quotients of the linear group.
In this talk, I will describe the second part of the proof that this result indeed holds. Our goal will be to explain a result, joint with Pierre Simon, on definable types in enrichments of NIP theories, which is crucial to prove elimination of imaginaries. We show that under certain hypothesis if a type in some NIP theory T is definable in an enrichment of T, then it h is already be definable in T.