As easy as $\mathbb {Q}$: Hilbert’s Tenth Problem for subrings of the rationals and number fields

Abstract

Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings $R$ of $\mathbb Q$ having the property that Hilbert's Tenth Problem for $R$, denoted $HTP(R)$, is Turing equivalent to $HTP(\mathbb Q)$. We are able to put several additional constraints on the rings $R$ that we construct. Given any computable nonnegative real number $r \leq 1$ we construct such a ring $R = Z[\frac1p : p \in S]$ with $S$ a set of primes of lower density $r$. We also construct examples of rings $R$ for which deciding membership in $R$ is Turing equivalent to deciding $HTP(R)$ and also equivalent to deciding $HTP(\mathbb Q)$. Alternatively, we can make $HTP(R)$ have arbitrary computably enumerable degree above $HTP(\mathbb Q)$. Finally, we show that the same can be done for subrings of number fields and their prime ideals.