### Abstract

We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K, a positive integer t > 1, and t nonnegative computable real numbers δ_{1}, . . . , δ_{t} whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S_{1}, . . . , S_{t} of densities δ_{1}, . . . , δ_{t}, respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring O _{K,Si}. We also show that we can find a partition as above such that each ring O_{K,Si} possesses an infinite Diophantine set which is discrete in every topology of the field. The only assumption on K we need is that there is an elliptic curve of rank one defined over K.

Original language | English (US) |
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Pages (from-to) | 1141-1162 |

Number of pages | 22 |

Journal | Mathematical Research Letters |

Volume | 18 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2011 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Mathematical Research Letters*,

*18*(6), 1141-1162. https://doi.org/10.4310/MRL.2011.v18.n6.a7

}

*Mathematical Research Letters*, vol. 18, no. 6, pp. 1141-1162. https://doi.org/10.4310/MRL.2011.v18.n6.a7

**Hilbert's tenth problem and Mazur's conjectures in complementary subrings of number fields.** / Eisenträger, Kirsten; Everest, Graham; Shlapentokh, Alexandra.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Hilbert's tenth problem and Mazur's conjectures in complementary subrings of number fields

AU - Eisenträger, Kirsten

AU - Everest, Graham

AU - Shlapentokh, Alexandra

PY - 2011/11

Y1 - 2011/11

N2 - We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K, a positive integer t > 1, and t nonnegative computable real numbers δ1, . . . , δt whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S1, . . . , St of densities δ1, . . . , δt, respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring O K,Si. We also show that we can find a partition as above such that each ring OK,Si possesses an infinite Diophantine set which is discrete in every topology of the field. The only assumption on K we need is that there is an elliptic curve of rank one defined over K.

AB - We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K, a positive integer t > 1, and t nonnegative computable real numbers δ1, . . . , δt whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S1, . . . , St of densities δ1, . . . , δt, respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring O K,Si. We also show that we can find a partition as above such that each ring OK,Si possesses an infinite Diophantine set which is discrete in every topology of the field. The only assumption on K we need is that there is an elliptic curve of rank one defined over K.

UR - http://www.scopus.com/inward/record.url?scp=84861054213&partnerID=8YFLogxK

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U2 - 10.4310/MRL.2011.v18.n6.a7

DO - 10.4310/MRL.2011.v18.n6.a7

M3 - Article

AN - SCOPUS:84861054213

VL - 18

SP - 1141

EP - 1162

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 6

ER -