TY - JOUR

T1 - As easy as ℚ

T2 - Hilbert’s tenth problem for subrings of the rationals and number fields

AU - Eisenträger, Kirsten

AU - Miller, Russell

AU - Park, Jennifer

AU - Shlapentokh, Alexandra

N1 - Funding Information:
Received by the editors February 9, 2016 and, in revised form, August 28, 2016 and September 22, 2016. 2010 Mathematics Subject Classification. Primary 11U05; Secondary 12L05, 03D45. The first author was partially supported by NSF grant DMS-1056703. The second author was partially supported by NSF grants DMS-1001306 and DMS-1362206 and by several PSC-CUNY Research Awards. The third author was partially supported by NSF grant DMS-1069236 and by an NSERC PDF grant. The fourth author was partially supported by NSF grant DMS-1161456.
Publisher Copyright:
© 2017 American Mathematical Society.

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

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U2 - 10.1090/tran/7075

DO - 10.1090/tran/7075

M3 - Article

AN - SCOPUS:85029916503

VL - 369

SP - 8291

EP - 8315

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 11

ER -