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Preprint Number 994

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994. Kirsten Eisentraeger, Russell Miller, Jennifer Park, and Alexandra Shlapentokh
As Easy as Q: Hilbert's Tenth Problem for Subrings of the Rationals and Number Fields

Submission date: 26 January 2016


Hilbert's Tenth Problem over the field 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 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 a ring R = Z[1/p : 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(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.

Mathematics Subject Classification: 11U05 (Primary) 12L05, 03D45 (Secondary)

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Full text arXiv 1601.07158: pdf, ps.

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