Interpretations of syntactic fragments of theories of fields

We set up general machinery to study interpretations of fragments of theories. We then apply this to existential fragments of theories of fields, and especially of henselian valued fields. As an application we prove many-one reductions between various existential theories of fields. In particular we exhibit several theories of fields many-one equivalent to the existential theory of $\mathbb{Q}$.
Comment: Results unchanged. Added explanations, in particular in Section 2. Slightly rearranged Section 3. To appear in Israel Journal of Mathematics