| Résume | In previous work with Fehm, and then Dittmann and Fehm, we found that the existential theory of an equicharacteristic henselian valued field is axiomatised using the existential theory of its residue field, conditionally, similar to an earlier theorem of Denef and Schoutens -- giving a transfer of decidability for existential theories. In this talk I’ll describe parts of ongoing work with Fehm (in the main different to those discussed recently at CIRM) in which we use an "abstract" framework for interpreting families of incomplete theories in others in order to find transfers of decidability in various settings. I will discuss consequences for theories of PAC fields and parts of the universal-existential theory of equicharacteristic henselian valued fields.
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