We prove model completeness for the theory of addition and the Frobenius map for certain subrings of rational functions in positive characteristic. More precisely: Let p be a prime number, F_p the prime field with p elements, F a field algebraic over F_p and z a variable. We show that the structures of rings R, which are generated over F[z] by adjoining a finite set of inverses of irreducible polynomials of F[z] (e.g., R = F_p[z, 1/z]), with addition, the Frobenius map x -> x^p and the predicate ‘in F’ - together with function symbols and constants that allow building all elements of F_p[z] - are model complete, i.e., each formula is equivalent to an existential formula. Further, we show that in these structures all questions, i.e., first order sentences, about the rings R may be, constructively, translated into questions about F.