{"paper":{"title":"Hilbert's Tenth Problem and Mazur's Conjectures in Complementary Subrings of Number Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.LO","authors_text":"Alexandra Shlapentokh, Graham Everest, Kirsten Eisentraeger","submitted_at":"2010-12-22T04:56:26Z","abstract_excerpt":"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 delta_1,..., delta_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 delta_1,..., delta_t, respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring O_{K,S_i}. We also "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.4878","kind":"arxiv","version":2},"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"}