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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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As easy as $\mathbb {Q}$: Hilbert’s Tenth Problem for subrings of the rationals and number fields
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by Kirsten Eisenträger, Russell Miller, Jennifer Park and Alexandra Shlapentokh PDF
Trans. Amer. Math. Soc. 369 (2017), 8291-8315 Request permission

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 $R \subseteq \mathbb {Q}$ having the property that Hilbert’s Tenth Problem for $R$, denoted $\operatorname {HTP}(R)$, is Turing equivalent to $\operatorname {HTP}(\mathbb {Q})$.

We are able to put several additional constraints on the rings $R$ that we construct. Given any computable nonnegative real number $r\leq 1$ we construct such rings $R=\mathbb {Z}[\mathcal {S}^{-1}]$ with $\mathcal {S}$ 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 $\operatorname {HTP}(R)$ and also equivalent to deciding $\operatorname {HTP}(\mathbb {Q})$. Alternatively, we can make $\operatorname {HTP}(R)$ have arbitrary computably enumerable degree above $\operatorname {HTP}(\mathbb {Q})$. Finally, we show that the same can be done for subrings of number fields and their prime ideals.

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Additional Information
  • Kirsten Eisenträger
  • Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
  • MR Author ID: 717302
  • Email: eisentra@math.psu.edu
  • Russell Miller
  • Affiliation: Department of Mathematics, Queens College, 65-30 Kissena Boulevard, Queens, New York 11367 – and – Ph.D. Programs in Mathematics and Computer Science, CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016
  • MR Author ID: 679194
  • Email: Russell.Miller@qc.cuny.edu
  • Jennifer Park
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
  • Email: jmypark@umich.edu
  • Alexandra Shlapentokh
  • Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
  • MR Author ID: 288363
  • ORCID: 0000-0003-1990-909X
  • Email: shlapentokha@ecu.edu
  • Received by editor(s): February 9, 2016
  • Received by editor(s) in revised form: August 28, 2016, and September 22, 2016
  • Published electronically: June 13, 2017
  • Additional Notes: 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.
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 8291-8315
  • MSC (2010): Primary 11U05; Secondary 12L05, 03D45
  • DOI: https://doi.org/10.1090/tran/7075
  • MathSciNet review: 3695862