{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:NQ2DRKPDUCHHX4PZM4WG2VTICF","short_pith_number":"pith:NQ2DRKPD","canonical_record":{"source":{"id":"1112.3474","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-12-15T10:45:31Z","cross_cats_sorted":[],"title_canon_sha256":"c2ee79b64a53c38e22d7d6aa1e5b4b497d3ba2eae4998843b59e50dec09a72e1","abstract_canon_sha256":"61112a2027de43edaac382e9b73b12c58a8368c05977e7a7be2dbc22009cdc0c"},"schema_version":"1.0"},"canonical_sha256":"6c3438a9e3a08e7bf1f9672c6d56681155620c8d3bf1ca2018a21e21cabc2c51","source":{"kind":"arxiv","id":"1112.3474","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.3474","created_at":"2026-05-18T03:57:40Z"},{"alias_kind":"arxiv_version","alias_value":"1112.3474v3","created_at":"2026-05-18T03:57:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3474","created_at":"2026-05-18T03:57:40Z"},{"alias_kind":"pith_short_12","alias_value":"NQ2DRKPDUCHH","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NQ2DRKPDUCHHX4PZ","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NQ2DRKPD","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:NQ2DRKPDUCHHX4PZM4WG2VTICF","target":"record","payload":{"canonical_record":{"source":{"id":"1112.3474","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-12-15T10:45:31Z","cross_cats_sorted":[],"title_canon_sha256":"c2ee79b64a53c38e22d7d6aa1e5b4b497d3ba2eae4998843b59e50dec09a72e1","abstract_canon_sha256":"61112a2027de43edaac382e9b73b12c58a8368c05977e7a7be2dbc22009cdc0c"},"schema_version":"1.0"},"canonical_sha256":"6c3438a9e3a08e7bf1f9672c6d56681155620c8d3bf1ca2018a21e21cabc2c51","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:40.361915Z","signature_b64":"xnwTEVdM5d98kVKC40O2mP+S8Y8ZrfOL4P7t5l5gNQIygrsZoe8CHuGpyDjoovUQ7dZbZk/0/R4FBI5PoPEVAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c3438a9e3a08e7bf1f9672c6d56681155620c8d3bf1ca2018a21e21cabc2c51","last_reissued_at":"2026-05-18T03:57:40.360960Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:40.360960Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.3474","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:57:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"drnIuOy+3TfkpIRuAXaaD4++v9zkd4WbwFFSmxvGyWK2mO4lsaNCH0vRJ1awUCu3OMdLT+jeBvRxDmDrLrNmBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:55:57.196993Z"},"content_sha256":"6dae53c11fc4d237c21571f7f02922ebf728ba42e8832c63457fb00d813724d1","schema_version":"1.0","event_id":"sha256:6dae53c11fc4d237c21571f7f02922ebf728ba42e8832c63457fb00d813724d1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:NQ2DRKPDUCHHX4PZM4WG2VTICF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Waring rank of the sum of pairwise coprime monomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Anthony V. Geramita, Enrico Carlini, Maria Virginia Catalisano","submitted_at":"2011-12-15T10:45:31Z","abstract_excerpt":"In this paper we compute the Waring rank of any polynomial of the form F=M_1+...+M_r, where the M_i are pairwise coprime monomials, i.e., GCD(M_i,M_j)=1 for i not j. In particular, we determine the Waring rank of any monomial. As an application we show that certain monomials in three variables give examples of forms of rank higher than the generic form. As a further application we produce a sum of power decomposition for any form which is the sum of pairwise coprime monomials."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3474","kind":"arxiv","version":3},"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":"2026-05-18T03:57:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g29EVq9fF47LWW3yuOSNh/2rt+mN1lgXkchMN1JUiiaOT6XHjSb41B43X6y5eEJ0jG+l2v0U0xygxbj67h+PDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:55:57.197397Z"},"content_sha256":"721dfdedad9b55a6a241a5a2734e23cfd1579effcdecafb6da365df0c0684a30","schema_version":"1.0","event_id":"sha256:721dfdedad9b55a6a241a5a2734e23cfd1579effcdecafb6da365df0c0684a30"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NQ2DRKPDUCHHX4PZM4WG2VTICF/bundle.json","state_url":"https://pith.science/pith/NQ2DRKPDUCHHX4PZM4WG2VTICF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NQ2DRKPDUCHHX4PZM4WG2VTICF/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":"2026-06-12T02:55:57Z","links":{"resolver":"https://pith.science/pith/NQ2DRKPDUCHHX4PZM4WG2VTICF","bundle":"https://pith.science/pith/NQ2DRKPDUCHHX4PZM4WG2VTICF/bundle.json","state":"https://pith.science/pith/NQ2DRKPDUCHHX4PZM4WG2VTICF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NQ2DRKPDUCHHX4PZM4WG2VTICF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:NQ2DRKPDUCHHX4PZM4WG2VTICF","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":"61112a2027de43edaac382e9b73b12c58a8368c05977e7a7be2dbc22009cdc0c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-12-15T10:45:31Z","title_canon_sha256":"c2ee79b64a53c38e22d7d6aa1e5b4b497d3ba2eae4998843b59e50dec09a72e1"},"schema_version":"1.0","source":{"id":"1112.3474","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.3474","created_at":"2026-05-18T03:57:40Z"},{"alias_kind":"arxiv_version","alias_value":"1112.3474v3","created_at":"2026-05-18T03:57:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3474","created_at":"2026-05-18T03:57:40Z"},{"alias_kind":"pith_short_12","alias_value":"NQ2DRKPDUCHH","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NQ2DRKPDUCHHX4PZ","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NQ2DRKPD","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:721dfdedad9b55a6a241a5a2734e23cfd1579effcdecafb6da365df0c0684a30","target":"graph","created_at":"2026-05-18T03:57:40Z","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":"In this paper we compute the Waring rank of any polynomial of the form F=M_1+...+M_r, where the M_i are pairwise coprime monomials, i.e., GCD(M_i,M_j)=1 for i not j. In particular, we determine the Waring rank of any monomial. As an application we show that certain monomials in three variables give examples of forms of rank higher than the generic form. As a further application we produce a sum of power decomposition for any form which is the sum of pairwise coprime monomials.","authors_text":"Anthony V. Geramita, Enrico Carlini, Maria Virginia Catalisano","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-12-15T10:45:31Z","title":"The Waring rank of the sum of pairwise coprime monomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3474","kind":"arxiv","version":3},"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":"sha256:6dae53c11fc4d237c21571f7f02922ebf728ba42e8832c63457fb00d813724d1","target":"record","created_at":"2026-05-18T03:57:40Z","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":"61112a2027de43edaac382e9b73b12c58a8368c05977e7a7be2dbc22009cdc0c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-12-15T10:45:31Z","title_canon_sha256":"c2ee79b64a53c38e22d7d6aa1e5b4b497d3ba2eae4998843b59e50dec09a72e1"},"schema_version":"1.0","source":{"id":"1112.3474","kind":"arxiv","version":3}},"canonical_sha256":"6c3438a9e3a08e7bf1f9672c6d56681155620c8d3bf1ca2018a21e21cabc2c51","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c3438a9e3a08e7bf1f9672c6d56681155620c8d3bf1ca2018a21e21cabc2c51","first_computed_at":"2026-05-18T03:57:40.360960Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:57:40.360960Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xnwTEVdM5d98kVKC40O2mP+S8Y8ZrfOL4P7t5l5gNQIygrsZoe8CHuGpyDjoovUQ7dZbZk/0/R4FBI5PoPEVAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:57:40.361915Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.3474","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6dae53c11fc4d237c21571f7f02922ebf728ba42e8832c63457fb00d813724d1","sha256:721dfdedad9b55a6a241a5a2734e23cfd1579effcdecafb6da365df0c0684a30"],"state_sha256":"4b3611dd064cc3105cfc96c7eca2551b6b23d834aef4186817d681e8176e169c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SIASwFUvbSNr/ddPPsw0nAzC0rKHeo+co3si4xTF9URY6ZdMl7XYKQkoEe8mGwBTX9TsnRfrmO17oNoV8IoZCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T02:55:57.199710Z","bundle_sha256":"08b2237d76e5d262a0bdb4f14c778bec00a3b01fbce10c1cc944f4c93c8c160b"}}