As easy as ℚ: Hilbert’s tenth problem for subrings of the rationals and number fields

Anne Kirsten Eisentraeger, Russell Miller, Jennifer Park, Alexandra Shlapentokh

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 languageEnglish (US)
Pages (from-to)8291-8315
Number of pages25
JournalTransactions of the American Mathematical Society
Volume369
Issue number11
DOIs
StatePublished - Jan 1 2017

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Number theory
Subring
Number field
Turing
Ring
Undecidability
Prime Ideal
Open Problems
Non-negative
Arbitrary

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

Eisentraeger, Anne Kirsten ; Miller, Russell ; Park, Jennifer ; Shlapentokh, Alexandra. / As easy as ℚ : Hilbert’s tenth problem for subrings of the rationals and number fields. In: Transactions of the American Mathematical Society. 2017 ; Vol. 369, No. 11. pp. 8291-8315.
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As easy as ℚ : Hilbert’s tenth problem for subrings of the rationals and number fields. / Eisentraeger, Anne Kirsten; Miller, Russell; Park, Jennifer; Shlapentokh, Alexandra.

In: Transactions of the American Mathematical Society, Vol. 369, No. 11, 01.01.2017, p. 8291-8315.

Research output: Contribution to journalArticle

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