Decidability of the theory of addition and the Frobenius map in rings of rational functions

Abstract

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 $F$.