{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:ERDBYAE4752RQS7ZHAWY3KQJSC","short_pith_number":"pith:ERDBYAE4","canonical_record":{"source":{"id":"1501.00671","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-04T12:56:47Z","cross_cats_sorted":[],"title_canon_sha256":"64f38bb0b6fa3e0be34b7391ccc76770d8e143ca039bf7edb8576c9995488f3f","abstract_canon_sha256":"5440c67217894a3fdc92be210b7f84415f8eefce0635cf840db2e607498616c4"},"schema_version":"1.0"},"canonical_sha256":"24461c009cff75184bf9382d8daa0990a60f087b03f02c6aa845f27b72362b9c","source":{"kind":"arxiv","id":"1501.00671","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.00671","created_at":"2026-05-18T02:30:02Z"},{"alias_kind":"arxiv_version","alias_value":"1501.00671v1","created_at":"2026-05-18T02:30:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00671","created_at":"2026-05-18T02:30:02Z"},{"alias_kind":"pith_short_12","alias_value":"ERDBYAE4752R","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"ERDBYAE4752RQS7Z","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"ERDBYAE4","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:ERDBYAE4752RQS7ZHAWY3KQJSC","target":"record","payload":{"canonical_record":{"source":{"id":"1501.00671","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-04T12:56:47Z","cross_cats_sorted":[],"title_canon_sha256":"64f38bb0b6fa3e0be34b7391ccc76770d8e143ca039bf7edb8576c9995488f3f","abstract_canon_sha256":"5440c67217894a3fdc92be210b7f84415f8eefce0635cf840db2e607498616c4"},"schema_version":"1.0"},"canonical_sha256":"24461c009cff75184bf9382d8daa0990a60f087b03f02c6aa845f27b72362b9c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:02.025608Z","signature_b64":"vKdhVGTNebAwJ7UxbelQBRaYc8yacKT3hs52Zyl++cuuksV6uDIctM0swqRIfap6KSUdvAXGIqk7w57Sx4OqAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24461c009cff75184bf9382d8daa0990a60f087b03f02c6aa845f27b72362b9c","last_reissued_at":"2026-05-18T02:30:02.025228Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:02.025228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.00671","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":"2026-05-18T02:30:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AGsQJi442FnmgTa60meGAAfz4Zcz2nCeO8tpmkLZgDpZrfXMG7/c1cNaeKPYGaRgNpj+plfhfDxZ5eg5z0zBBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T06:06:45.014676Z"},"content_sha256":"e739c980535f9b0fe8fc2a51611c6b2c5be9dfe596a3affed9eb54e41c4fa9d3","schema_version":"1.0","event_id":"sha256:e739c980535f9b0fe8fc2a51611c6b2c5be9dfe596a3affed9eb54e41c4fa9d3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:ERDBYAE4752RQS7ZHAWY3KQJSC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Complexity of Noncommutative Polynomial Factorization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Gaurav Rattan, Pushkar S Joglekar, V. Arvind","submitted_at":"2015-01-04T12:56:47Z","abstract_excerpt":"In this paper we study the complexity of factorization of polynomials in the free noncommutative ring $\\mathbb{F}\\langle x_1,x_2,\\dots,x_n\\rangle$ of polynomials over the field $\\mathbb{F}$ and noncommuting variables $x_1,x_2,\\ldots,x_n$. Our main results are the following.\n  Although $\\mathbb{F}\\langle x_1,x_2,\\dots,x_n \\rangle$ is not a unique factorization ring, we note that variable-disjoint factorization in $\\mathbb{F}\\langle x_1,x_2,\\dots,x_n \\rangle$ has the uniqueness property. Furthermore, we prove that computing the variable-disjoint factorization is polynomial-time equivalent to Pol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00671","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":"2026-05-18T02:30:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D1VpUFzCbAGNlEzZw7uS0not/sV6w+H3ij0L/tZvyt7rK3kVzu+e38/eDs2YXx5yrIvFnX1SZFZvSdW/8Z0eDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T06:06:45.015387Z"},"content_sha256":"ff557ca90355bb8582082486f08c218339202ea83715b5d961827168881c9688","schema_version":"1.0","event_id":"sha256:ff557ca90355bb8582082486f08c218339202ea83715b5d961827168881c9688"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ERDBYAE4752RQS7ZHAWY3KQJSC/bundle.json","state_url":"https://pith.science/pith/ERDBYAE4752RQS7ZHAWY3KQJSC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ERDBYAE4752RQS7ZHAWY3KQJSC/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":"2026-06-09T06:06:45Z","links":{"resolver":"https://pith.science/pith/ERDBYAE4752RQS7ZHAWY3KQJSC","bundle":"https://pith.science/pith/ERDBYAE4752RQS7ZHAWY3KQJSC/bundle.json","state":"https://pith.science/pith/ERDBYAE4752RQS7ZHAWY3KQJSC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ERDBYAE4752RQS7ZHAWY3KQJSC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ERDBYAE4752RQS7ZHAWY3KQJSC","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":"5440c67217894a3fdc92be210b7f84415f8eefce0635cf840db2e607498616c4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-04T12:56:47Z","title_canon_sha256":"64f38bb0b6fa3e0be34b7391ccc76770d8e143ca039bf7edb8576c9995488f3f"},"schema_version":"1.0","source":{"id":"1501.00671","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.00671","created_at":"2026-05-18T02:30:02Z"},{"alias_kind":"arxiv_version","alias_value":"1501.00671v1","created_at":"2026-05-18T02:30:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00671","created_at":"2026-05-18T02:30:02Z"},{"alias_kind":"pith_short_12","alias_value":"ERDBYAE4752R","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"ERDBYAE4752RQS7Z","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"ERDBYAE4","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:ff557ca90355bb8582082486f08c218339202ea83715b5d961827168881c9688","target":"graph","created_at":"2026-05-18T02:30:02Z","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 study the complexity of factorization of polynomials in the free noncommutative ring $\\mathbb{F}\\langle x_1,x_2,\\dots,x_n\\rangle$ of polynomials over the field $\\mathbb{F}$ and noncommuting variables $x_1,x_2,\\ldots,x_n$. Our main results are the following.\n  Although $\\mathbb{F}\\langle x_1,x_2,\\dots,x_n \\rangle$ is not a unique factorization ring, we note that variable-disjoint factorization in $\\mathbb{F}\\langle x_1,x_2,\\dots,x_n \\rangle$ has the uniqueness property. Furthermore, we prove that computing the variable-disjoint factorization is polynomial-time equivalent to Pol","authors_text":"Gaurav Rattan, Pushkar S Joglekar, V. Arvind","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-04T12:56:47Z","title":"On the Complexity of Noncommutative Polynomial Factorization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00671","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:e739c980535f9b0fe8fc2a51611c6b2c5be9dfe596a3affed9eb54e41c4fa9d3","target":"record","created_at":"2026-05-18T02:30:02Z","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":"5440c67217894a3fdc92be210b7f84415f8eefce0635cf840db2e607498616c4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-01-04T12:56:47Z","title_canon_sha256":"64f38bb0b6fa3e0be34b7391ccc76770d8e143ca039bf7edb8576c9995488f3f"},"schema_version":"1.0","source":{"id":"1501.00671","kind":"arxiv","version":1}},"canonical_sha256":"24461c009cff75184bf9382d8daa0990a60f087b03f02c6aa845f27b72362b9c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"24461c009cff75184bf9382d8daa0990a60f087b03f02c6aa845f27b72362b9c","first_computed_at":"2026-05-18T02:30:02.025228Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:30:02.025228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vKdhVGTNebAwJ7UxbelQBRaYc8yacKT3hs52Zyl++cuuksV6uDIctM0swqRIfap6KSUdvAXGIqk7w57Sx4OqAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:30:02.025608Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.00671","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e739c980535f9b0fe8fc2a51611c6b2c5be9dfe596a3affed9eb54e41c4fa9d3","sha256:ff557ca90355bb8582082486f08c218339202ea83715b5d961827168881c9688"],"state_sha256":"0ca5252bcc510e82e758d06a29b814fc7d8016897960b0b9e9f7778eafd32de9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E3RHDbNjOc8I9f62p0+HYOA9SbmlQShNpa7F8xKNE0SFsV+GU/jtEc2ip0ssm5HfGIvdV4B9Jyefy88Q8ygQBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T06:06:45.019078Z","bundle_sha256":"220a0a6be6a27e02709821cdbd52732c064ae58caa93f09ee7c0e37c24f32168"}}