Existential decidability for addition and divisibility in holomorphy subrings of global fields

TL;DR

This paper explores the decidability of solving linear equations with divisibility constraints over holomorphy subrings of global fields, revealing conditions for both decidability and undecidability based on pole locations.

Contribution

It provides new decidability and undecidability results for systems of linear equations with divisibility over holomorphy subrings, depending on the set of places where poles are allowed.

Findings

Decidability when poles are at a cofinite set of places

Undecidability when poles are at a finite set of places

Conditions for solving linear systems with divisibility in holomorphy subrings

Abstract

We investigate the problem of deciding whether a system of linear equations, together with divisibility conditions on the variables, has a solution over holomorphy subrings of global fields. We obtain decidability results when we allow poles at a cofinite set of places, and undecidability results when at a finite set of places.