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By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\\in K[X]$ maps every element of $O_K$ of degree $n$ to an algebraic integer, then $h(X)$ is integral-valued over $O_K$, that is $h(O_K)\\subset O_K$. A similar property holds if we consider the set of all algebraic integers of degree $n$ and a polynomial $f\\in\\mathbb{Q}[X]$: if $f(\\alpha)$ is integral over $\\mathbb{Z}$ for every algebraic integer $\\alpha$ of degree $n$, then $f(\\beta)$ is integral over"},"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":"1301.2045","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-01-10T07:58:37Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"36b013a0086f18f02541242916abae557ed1ecf37b33f2a5ad5db9f9a784ae27","abstract_canon_sha256":"55b74e60da82cd1da4f18bf1f26af21d6cad7b49d0a4f391d3012e98dc1ceb77"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:24.583642Z","signature_b64":"uzOF2lXBY9pdUr0z+y7t3b9qtOyCgvkUbqoiJwS0xpekhovPrdMTr8cq/DoSIH+LgUq3pxLCwPnTdEGbpQKABQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8516971f6cdbb741c66f3568662c34898c6ec2ef405fad8505e24ad6e9ab7337","last_reissued_at":"2026-05-18T00:04:24.583088Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:24.583088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Integral-valued polynomials over the set of algebraic integers of bounded degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.NT","authors_text":"Giulio Peruginelli","submitted_at":"2013-01-10T07:58:37Z","abstract_excerpt":"Let $K$ be a number field of degree $n$ with ring of integers $O_K$. 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