{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:7FWJQEZAETS6JMBWXNJ3NJJL4E","short_pith_number":"pith:7FWJQEZA","canonical_record":{"source":{"id":"1008.1286","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-06T22:13:54Z","cross_cats_sorted":[],"title_canon_sha256":"414308b54b50b33f45300cc785787a096b3a256373cfc76d824de4c5f6f0b490","abstract_canon_sha256":"fbacb31bc157d9e168f10c5b07e5a1128887f5203348501192e47e4357b4fb5f"},"schema_version":"1.0"},"canonical_sha256":"f96c98132024e5e4b036bb53b6a52be11bd897f7e32ccf0238eff30569df08d5","source":{"kind":"arxiv","id":"1008.1286","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.1286","created_at":"2026-05-18T04:42:33Z"},{"alias_kind":"arxiv_version","alias_value":"1008.1286v1","created_at":"2026-05-18T04:42:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1286","created_at":"2026-05-18T04:42:33Z"},{"alias_kind":"pith_short_12","alias_value":"7FWJQEZAETS6","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"7FWJQEZAETS6JMBW","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"7FWJQEZA","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:7FWJQEZAETS6JMBWXNJ3NJJL4E","target":"record","payload":{"canonical_record":{"source":{"id":"1008.1286","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-06T22:13:54Z","cross_cats_sorted":[],"title_canon_sha256":"414308b54b50b33f45300cc785787a096b3a256373cfc76d824de4c5f6f0b490","abstract_canon_sha256":"fbacb31bc157d9e168f10c5b07e5a1128887f5203348501192e47e4357b4fb5f"},"schema_version":"1.0"},"canonical_sha256":"f96c98132024e5e4b036bb53b6a52be11bd897f7e32ccf0238eff30569df08d5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:33.404310Z","signature_b64":"xII+PEmoWmCWhB0eDs2zuAsbU4lOlGB0Q6pIuuO0lVdhane3F9rRzL0akJrUACvKgp/ZCq1zvs0hlQAEJhexAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f96c98132024e5e4b036bb53b6a52be11bd897f7e32ccf0238eff30569df08d5","last_reissued_at":"2026-05-18T04:42:33.403597Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:33.403597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.1286","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":"2026-05-18T04:42:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cAwDxVFFOWnxzoD7P5DJAyZVkD13Ru67VDzcNM7F1gLm838Unz1u1MXcGamaa/fztgSXe+9aF5NddN9REDeWCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:34:00.628837Z"},"content_sha256":"a9a5f9336958cc1edb11786a7da00296bc95a3a2f61af71b211a9c819a152ddd","schema_version":"1.0","event_id":"sha256:a9a5f9336958cc1edb11786a7da00296bc95a3a2f61af71b211a9c819a152ddd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:7FWJQEZAETS6JMBWXNJ3NJJL4E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Subalgebras of Matrix Algebras Generated by Companion Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Fernando Szechtman, Natalio H. Guersenzvaig","submitted_at":"2010-08-06T22:13:54Z","abstract_excerpt":"Let $f,g\\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in the full integral lattice $M_n(Z)$, in which case we compute the exact value of this index in terms of the resultant of $f$ and $g$. If $R$ is a commutative ring with identity we determine when $R< C,D>=M_n(R)$, in which case a presentation for $M_n(R)$ in terms of $C$ and $D$ is given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1286","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":"2026-05-18T04:42:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xBW47zrfNquIO1po0VKxEiau+iOVEmk4mpzCwPryPqaICkBfVNRnFM/s06cMyIwPDu33ESjlNMSJQWVPGxpYBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:34:00.629183Z"},"content_sha256":"18afd5b3b61c9b048a756fdc73f63027d9562f41f18c59077ed1895f36e997e8","schema_version":"1.0","event_id":"sha256:18afd5b3b61c9b048a756fdc73f63027d9562f41f18c59077ed1895f36e997e8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7FWJQEZAETS6JMBWXNJ3NJJL4E/bundle.json","state_url":"https://pith.science/pith/7FWJQEZAETS6JMBWXNJ3NJJL4E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7FWJQEZAETS6JMBWXNJ3NJJL4E/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":"2026-06-24T21:34:00Z","links":{"resolver":"https://pith.science/pith/7FWJQEZAETS6JMBWXNJ3NJJL4E","bundle":"https://pith.science/pith/7FWJQEZAETS6JMBWXNJ3NJJL4E/bundle.json","state":"https://pith.science/pith/7FWJQEZAETS6JMBWXNJ3NJJL4E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7FWJQEZAETS6JMBWXNJ3NJJL4E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7FWJQEZAETS6JMBWXNJ3NJJL4E","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":"fbacb31bc157d9e168f10c5b07e5a1128887f5203348501192e47e4357b4fb5f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-06T22:13:54Z","title_canon_sha256":"414308b54b50b33f45300cc785787a096b3a256373cfc76d824de4c5f6f0b490"},"schema_version":"1.0","source":{"id":"1008.1286","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.1286","created_at":"2026-05-18T04:42:33Z"},{"alias_kind":"arxiv_version","alias_value":"1008.1286v1","created_at":"2026-05-18T04:42:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1286","created_at":"2026-05-18T04:42:33Z"},{"alias_kind":"pith_short_12","alias_value":"7FWJQEZAETS6","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"7FWJQEZAETS6JMBW","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"7FWJQEZA","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:18afd5b3b61c9b048a756fdc73f63027d9562f41f18c59077ed1895f36e997e8","target":"graph","created_at":"2026-05-18T04:42:33Z","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 $f,g\\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in the full integral lattice $M_n(Z)$, in which case we compute the exact value of this index in terms of the resultant of $f$ and $g$. If $R$ is a commutative ring with identity we determine when $R< C,D>=M_n(R)$, in which case a presentation for $M_n(R)$ in terms of $C$ and $D$ is given.","authors_text":"Fernando Szechtman, Natalio H. Guersenzvaig","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-06T22:13:54Z","title":"Subalgebras of Matrix Algebras Generated by Companion Matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1286","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:a9a5f9336958cc1edb11786a7da00296bc95a3a2f61af71b211a9c819a152ddd","target":"record","created_at":"2026-05-18T04:42:33Z","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":"fbacb31bc157d9e168f10c5b07e5a1128887f5203348501192e47e4357b4fb5f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-08-06T22:13:54Z","title_canon_sha256":"414308b54b50b33f45300cc785787a096b3a256373cfc76d824de4c5f6f0b490"},"schema_version":"1.0","source":{"id":"1008.1286","kind":"arxiv","version":1}},"canonical_sha256":"f96c98132024e5e4b036bb53b6a52be11bd897f7e32ccf0238eff30569df08d5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f96c98132024e5e4b036bb53b6a52be11bd897f7e32ccf0238eff30569df08d5","first_computed_at":"2026-05-18T04:42:33.403597Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:33.403597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xII+PEmoWmCWhB0eDs2zuAsbU4lOlGB0Q6pIuuO0lVdhane3F9rRzL0akJrUACvKgp/ZCq1zvs0hlQAEJhexAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:33.404310Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.1286","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a9a5f9336958cc1edb11786a7da00296bc95a3a2f61af71b211a9c819a152ddd","sha256:18afd5b3b61c9b048a756fdc73f63027d9562f41f18c59077ed1895f36e997e8"],"state_sha256":"74df51fb9c13ff34fc0b60f7b98258d65e07aa4b1dd75f4b6db554a067bbe4af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q0i6vnsttqlNODZHwIEVIhLpaQeCCSBsxH/nRTgwez5e+QtDaNTzRcTBperlsuYaxRpJHnGVRw8S9K4xu99NBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T21:34:00.631166Z","bundle_sha256":"7aab02b2aa3e497f0388c34989c72175823c6435165e1feb215d04b89eef1b7d"}}