The Diophantine Problem for Addition and Divisibility Over Subrings of the Rationals

Abstract

It is shown that the positive existential theory of the structure (ℤ[S−1]; =, 0, 1, + , |), where S is a nonempty finite set of prime numbers, is undecidable. This result should be put in contrast with the fact that the positive existential theory of (ℤ; =, 0, 1, + |) is decidable.

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

“…Complementing the above result of Lipshitz and Beltyukov, it was recently shown by Cerda-Romero and Martínez-Ranero [CM17] that if S is a finite and non-empty set of primes, then there is no algorithm to decide whether Equation (1) has solutions in Z[S −1 ]. There are also similar results for rings of rational functions over finite fields see [Ph85], [Ph88] and [CM20].…”

“…The formulas we use are quite similar to the ones used in [Lip78b] and [CM17]. However, the proofs are quite different, the main new ingredients are the use of the pigeon hole principle (which allows us to improve the result of Lipshitz) and the use of the ideal class number and the lattice of units of O × K,S (which allows us to extend the results of [CM17]). The structure of the paper is as follows.…”