{"paper":{"title":"As Easy as $\\mathbb Q$: Hilbert's Tenth Problem for Subrings of the Rationals and Number Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.NT","authors_text":"Alexandra Shlapentokh, Jennifer Park, Kirsten Eisentraeger, Russell Miller","submitted_at":"2016-01-26T20:38:39Z","abstract_excerpt":"Hilbert's Tenth Problem over the field $\\mathbb Q$ of rational numbers is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings $R$ of $\\mathbb Q$ having the property that Hilbert's Tenth Problem for $R$, denoted $HTP(R)$, is Turing equivalent to $HTP(\\mathbb Q)$.\n  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 a ring $R = Z[\\frac1p : p \\in S]$ with $S$ a set of primes of lower density $r$. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07158","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}