### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 8291-8315 |

Number of pages | 25 |

Journal | Transactions of the American Mathematical Society |

Volume | 369 |

Issue number | 11 |

DOIs | |

State | Published - Jan 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*369*(11), 8291-8315. https://doi.org/10.1090/tran/7075

}

*Transactions of the American Mathematical Society*, vol. 369, no. 11, pp. 8291-8315. https://doi.org/10.1090/tran/7075

**As easy as ℚ : Hilbert’s tenth problem for subrings of the rationals and number fields.** / Eisenträger, Kirsten; Miller, Russell; Park, Jennifer; Shlapentokh, Alexandra.

Research output: Contribution to journal › Article

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

PY - 2017/1/1

Y1 - 2017/1/1

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.

UR - http://www.scopus.com/inward/record.url?scp=85029916503&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85029916503&partnerID=8YFLogxK

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 -