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Write $S$ for the maximum of $v(\\gcd (f(n), g(n)))$ over all integers $n$. It is known that $S \\le v(r)$. We give various lower and upper bounds for the least possible value of $v(r)-S$ provided that a given power $p^s$ divides both $f(n)$ and $g(n)$ for all $n$. In particular, the least possible value is $ps^2-s$ for $s\\le p$ and is asymptotically $(p-1)s^2$ for large $s$."},"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":"1712.01054","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-12-04T13:10:38Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"89e8880f3ebae9e7278bcf1c6e49899325649e16517f7b23c814c3881f4ecda6","abstract_canon_sha256":"a22e4d53c14009da066b64dcf9e233f84f199c08587ba89da4abcf7061d7ec18"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:45.252639Z","signature_b64":"vblEVy1tFry8VzZLxk/yPQNMRjwVI9CnIGulJsVOZIylTvYHzTNRpJatqgx8cvXVvOz/NHS5f7MqUzYbNiCmDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2592738b6a3c689c7b19c332d270d74aca48a4ba0bc6d4c0f8af3e7897ce0b7f","last_reissued_at":"2026-05-18T00:13:45.251918Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:45.251918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Estimating the greatest common divisor of the value of two polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.NT","authors_text":"Gergely Z\\'abr\\'adi, P\\'eter E. Frenkel","submitted_at":"2017-12-04T13:10:38Z","abstract_excerpt":"Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of $v(\\gcd (f(n), g(n)))$ over all integers $n$. It is known that $S \\le v(r)$. We give various lower and upper bounds for the least possible value of $v(r)-S$ provided that a given power $p^s$ divides both $f(n)$ and $g(n)$ for all $n$. 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