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

TL;DR

This paper proves that certain rings of rational functions in positive characteristic have a model complete theory when considering addition, the Frobenius map, and a predicate for the base field, enabling reduction of questions to the base field.

Contribution

It establishes model completeness for the theory of addition, Frobenius map, and a predicate in specific subrings of rational functions over finite fields.

Findings

Structures are model complete, with all formulas equivalent to existential ones.

First-order questions about these rings can be translated into questions about the base field.

The results apply to rings generated over $F[z]$ by inverting finitely many irreducible polynomials.

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 $F$.