Stably embedded submodels of Henselian valued fields

Abstract: We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued fields is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields satisfies a certain algebraic condition. We show for instance that all types over the Hahn field $\mathbb{R}((\mathbb{Z}))$ are definable. Similarly, all types over the quotient field of the Witt ring $W(\mathbb{F}_p^{\text{alg}})$ are definable. This extends a work of Cubides and Delon and of Cubides and Ye.