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

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.

We improved the statement of Theorem 1.2 -- we thank S. Anscombe and A. Fehm for this