{"paper":{"title":"Comments on the height reducing property II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"J\\\"org M. Thuswaldner, Shigeki Akiyama, Toufik Za\\\"imi","submitted_at":"2014-03-28T18:36:27Z","abstract_excerpt":"A complex number $\\alpha$ is said to satisfy the height reducing property if there is a finite set $F\\subset \\mathbb{Z}$ such that $\\mathbb{Z}[\\alpha]=F[\\alpha]$, where $\\mathbb{Z}$ is the ring of the rational integers. It is easy to see that $\\alpha$ is an algebraic number when it satisfies the height reducing property. We prove the relation $\\operatorname{Card}(F)\\geq \\max\\{2,\\left\\vert M_{\\alpha}(0)\\right\\vert \\},$ where $M_{\\alpha}$ is the minimal polynomial of $\\alpha$ over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers $\\al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7480","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"}