Algebraic points in tame expansions of fields

We investigate the behaviour of algebraic points in several expansions of the real, complex and p-adic fields. We build off work of Eleftheriou, Günaydin and Hieronymi to prove a Pila-Wilkie result for a p-adic subanalytic structure with a predicate for either a dense elementary substructure or a dense dcl-independent set. In the process we prove a structure theorem for p-minimal structures with a predicate for a dense independent set. We then prove quantifier reduction results for the complex field with a predicate for the singular moduli and the real field with an exponentially transcendental power function and a predicate for the algebraic numbers using a Schanuel property proved by Bays, Kirby and Wilkie. Finally we adapt a theorem by Ax about exponential fields, key to the proof of the Schanuel property for power functions, to power functions.