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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1403.7480","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-03-28T18:36:27Z","cross_cats_sorted":[],"title_canon_sha256":"7f5498903300fb695a5a3af638b28649d4eac4d078bbe9a601292222fe03d9e8","abstract_canon_sha256":"c1da95617868198bdeeed011793b780cca8ad492b719d74098d1280dcd21386b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:55.173173Z","signature_b64":"zEPNLstw45qVVCQgHSJRQrH/dxE9i1bJRjH0tmgNq0In1NAbIT9+kGZ+K4JQV3LD8MpWt6O6+jkwMgKL8q2ICA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f314433fb4a00c6031e7476f6d57f80e448a5277f5d563fe3b47fee6cd8d1286","last_reissued_at":"2026-05-18T02:28:55.172706Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:55.172706Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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