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

Kirsten Eisenträger, Graham Everest, Alexandra Shlapentokh

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 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.

Original languageEnglish (US)
Pages (from-to)1141-1162
Number of pages22
JournalMathematical Research Letters
Volume18
Issue number6
DOIs
StatePublished - Nov 2011

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Hilbert's Tenth Problem
Subring
Number field
Ring
P-adic
Elliptic Curves
Disjoint
Non-negative
Partition
Topology
Integer
Subset

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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

In: Mathematical Research Letters, Vol. 18, No. 6, 11.2011, p. 1141-1162.

Research output: Contribution to journalArticle

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