As Easy as Q: Hilbert’s Tenth Problem for Subrings of the Rationals and Number Fields

Hilbert’s Tenth Problem over the rationals 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 ⊆ Q having the property that Hilbert’s Tenth Problem for R, denoted HTP(R), is Turing equivalent to HTP(Q). We are able to put several additional constraints on the rings R that we construct. Given any computable nonnegative real number r ≤ 1 we construct such rings R = Z[S −1] 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(Q). Alternatively, we can make HTP(R) have arbitrary computably enumerable degree above HTP(Q). Finally, we show that the same can be done for subrings of number fields and their prime ideals.