  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, $\mathbb{F}_{p}$ the prime field with $p$
elements, $F$ a field algebraic over $\mathbb{F}_p$ 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.,
$R=\mathbb{F}_p[z, \frac{1}{z}]$), with addition, the Frobenius map $x\mapsto
x^p$ and the predicate '$\in F$' - together with function symbols and constants
that allow building all elements of $\mathbb{F}_p[z]$ - are model complete,
i.e., each formula is equivalent to an existential formula. Further, we show
that in these structures all questions, i.e., \emph{first order sentences},
about the rings $R$ may be, constructively, translated into questions about
$F$.
