{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:FX3BPCDF5CAMG2QXS7S33EWKIE","short_pith_number":"pith:FX3BPCDF","canonical_record":{"source":{"id":"1012.3897","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-17T15:00:05Z","cross_cats_sorted":[],"title_canon_sha256":"3e4bf3336d390eff96c161643ff3fe213676f0299aa6401ec3c492d5ddb459ac","abstract_canon_sha256":"966d678f4e6b05cb704fb45f84af9588267fcad9ced6df9e67877fe749cfe3b5"},"schema_version":"1.0"},"canonical_sha256":"2df6178865e880c36a1797e5bd92ca412ce387dbfef9ccd7cc045c73f086e69c","source":{"kind":"arxiv","id":"1012.3897","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.3897","created_at":"2026-05-18T03:51:58Z"},{"alias_kind":"arxiv_version","alias_value":"1012.3897v1","created_at":"2026-05-18T03:51:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3897","created_at":"2026-05-18T03:51:58Z"},{"alias_kind":"pith_short_12","alias_value":"FX3BPCDF5CAM","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FX3BPCDF5CAMG2QX","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FX3BPCDF","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:FX3BPCDF5CAMG2QXS7S33EWKIE","target":"record","payload":{"canonical_record":{"source":{"id":"1012.3897","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-17T15:00:05Z","cross_cats_sorted":[],"title_canon_sha256":"3e4bf3336d390eff96c161643ff3fe213676f0299aa6401ec3c492d5ddb459ac","abstract_canon_sha256":"966d678f4e6b05cb704fb45f84af9588267fcad9ced6df9e67877fe749cfe3b5"},"schema_version":"1.0"},"canonical_sha256":"2df6178865e880c36a1797e5bd92ca412ce387dbfef9ccd7cc045c73f086e69c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:58.761996Z","signature_b64":"4JXrn1vfIzZ1PB+3oOV8JNZejGMLfGmrnzSIRrDBu+lBqwMA0HFtXQbThgSsUbDahGjJVXqiWM5X3T62Rg/KCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2df6178865e880c36a1797e5bd92ca412ce387dbfef9ccd7cc045c73f086e69c","last_reissued_at":"2026-05-18T03:51:58.761250Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:58.761250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.3897","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:51:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OEsnPq6m8Sbk1GrOKYjNNMzVpAJnn5p4xqngzxIlhJHpEjxNRlPxzUi2nSBrX3NeBQTz952Q9mxbojII9tXYAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T18:44:22.710018Z"},"content_sha256":"23d491e8ab3091b4420d512a9c68be5110af87da7ceb657fdf315988b48eb692","schema_version":"1.0","event_id":"sha256:23d491e8ab3091b4420d512a9c68be5110af87da7ceb657fdf315988b48eb692"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:FX3BPCDF5CAMG2QXS7S33EWKIE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the height of cyclotomic polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bartlomiej Bzdega","submitted_at":"2010-12-17T15:00:05Z","abstract_excerpt":"Let $A_n$ denote the height of cyclotomic polynomial $\\Phi_n$, where $n$ is a product of $k$ distinct odd primes. We prove that $A_n \\le \\epsilon_k\\phi(n)^{k^{-1}2^{k-1}-1}$ with $-\\log\\epsilon_k\\sim c2^k$, $c>0$. The same statement is true for the height $C_n$ of the inverse cyclotomic polynomial $\\Psi_n$.\n  Additionally, we improve on a bound of Kaplan for the maximal height of divisors of $x^n-1$, denoted by $B_n$. We show that $B_n<\\eta_k n^{(3^k-1)/(2k)-1}$, with $-\\log \\eta_k \\sim c3^k$ and the same $c$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3897","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:51:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SGZQDS7natEFmq0AfD7rJ9j7u3o/Gy9CbulSXKD+TFC3tOhRiWNgUxABvaEUw8Q03FdqDwNi/F7ANWofRuTLDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T18:44:22.710383Z"},"content_sha256":"eb1496bc8ddff1e64996811b1502ca7422a64a05a91591e64991d66f6ac00cc5","schema_version":"1.0","event_id":"sha256:eb1496bc8ddff1e64996811b1502ca7422a64a05a91591e64991d66f6ac00cc5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FX3BPCDF5CAMG2QXS7S33EWKIE/bundle.json","state_url":"https://pith.science/pith/FX3BPCDF5CAMG2QXS7S33EWKIE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FX3BPCDF5CAMG2QXS7S33EWKIE/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-30T18:44:22Z","links":{"resolver":"https://pith.science/pith/FX3BPCDF5CAMG2QXS7S33EWKIE","bundle":"https://pith.science/pith/FX3BPCDF5CAMG2QXS7S33EWKIE/bundle.json","state":"https://pith.science/pith/FX3BPCDF5CAMG2QXS7S33EWKIE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FX3BPCDF5CAMG2QXS7S33EWKIE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:FX3BPCDF5CAMG2QXS7S33EWKIE","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"966d678f4e6b05cb704fb45f84af9588267fcad9ced6df9e67877fe749cfe3b5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-17T15:00:05Z","title_canon_sha256":"3e4bf3336d390eff96c161643ff3fe213676f0299aa6401ec3c492d5ddb459ac"},"schema_version":"1.0","source":{"id":"1012.3897","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.3897","created_at":"2026-05-18T03:51:58Z"},{"alias_kind":"arxiv_version","alias_value":"1012.3897v1","created_at":"2026-05-18T03:51:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3897","created_at":"2026-05-18T03:51:58Z"},{"alias_kind":"pith_short_12","alias_value":"FX3BPCDF5CAM","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FX3BPCDF5CAMG2QX","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FX3BPCDF","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:eb1496bc8ddff1e64996811b1502ca7422a64a05a91591e64991d66f6ac00cc5","target":"graph","created_at":"2026-05-18T03:51:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $A_n$ denote the height of cyclotomic polynomial $\\Phi_n$, where $n$ is a product of $k$ distinct odd primes. We prove that $A_n \\le \\epsilon_k\\phi(n)^{k^{-1}2^{k-1}-1}$ with $-\\log\\epsilon_k\\sim c2^k$, $c>0$. The same statement is true for the height $C_n$ of the inverse cyclotomic polynomial $\\Psi_n$.\n  Additionally, we improve on a bound of Kaplan for the maximal height of divisors of $x^n-1$, denoted by $B_n$. We show that $B_n<\\eta_k n^{(3^k-1)/(2k)-1}$, with $-\\log \\eta_k \\sim c3^k$ and the same $c$.","authors_text":"Bartlomiej Bzdega","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-17T15:00:05Z","title":"On the height of cyclotomic polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3897","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:23d491e8ab3091b4420d512a9c68be5110af87da7ceb657fdf315988b48eb692","target":"record","created_at":"2026-05-18T03:51:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"966d678f4e6b05cb704fb45f84af9588267fcad9ced6df9e67877fe749cfe3b5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-17T15:00:05Z","title_canon_sha256":"3e4bf3336d390eff96c161643ff3fe213676f0299aa6401ec3c492d5ddb459ac"},"schema_version":"1.0","source":{"id":"1012.3897","kind":"arxiv","version":1}},"canonical_sha256":"2df6178865e880c36a1797e5bd92ca412ce387dbfef9ccd7cc045c73f086e69c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2df6178865e880c36a1797e5bd92ca412ce387dbfef9ccd7cc045c73f086e69c","first_computed_at":"2026-05-18T03:51:58.761250Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:51:58.761250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4JXrn1vfIzZ1PB+3oOV8JNZejGMLfGmrnzSIRrDBu+lBqwMA0HFtXQbThgSsUbDahGjJVXqiWM5X3T62Rg/KCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:51:58.761996Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.3897","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:23d491e8ab3091b4420d512a9c68be5110af87da7ceb657fdf315988b48eb692","sha256:eb1496bc8ddff1e64996811b1502ca7422a64a05a91591e64991d66f6ac00cc5"],"state_sha256":"e9ab9c209e175eeaee6e17b13d8addb101378905d85689b5127d70d28aa333eb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Cz6WlQOWr6AFz4Gn9gsO5q1QjR6yHUqfiWvUTBUrk9L28ckVlpOPY9iyQVhWPBk4UjgNmmkUzpUVnc8nQ2w2Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T18:44:22.712728Z","bundle_sha256":"0418c254527c950db8a1691500130b1bed946ff79bbf2fe067877973bd54f429"}}