{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:1994:B2P6QCE332DZMHJRQDYN7YF3FY","short_pith_number":"pith:B2P6QCE3","schema_version":"1.0","canonical_sha256":"0e9fe8089bde87961d3180f0dfe0bb2e08991d3975ae8d6fa24df29c601082ae","source":{"kind":"arxiv","id":"math/9410207","version":1},"attestation_state":"computed","paper":{"title":"Matrix Vieta Theorem","license":"","headline":"","cross_cats":["math.OA"],"primary_cat":"math.RA","authors_text":"Albert Schwarz, Dmitry Fuchs","submitted_at":"1994-10-25T00:00:00Z","abstract_excerpt":"We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-$k$ matrices.\n  Specifically, we prove that if $X_1,\\dots ,X_n$ are solutions of an algebraic matrix equation $X^n+A_1X^{n-1}+\\dots +A_n=0,$ independent in the sense that they determine the coefficients $A_1,\\dots ,A_n$, then the trace of $A_1$ is the sum of the traces of the $X_i$, and the determinant of $A_n$ is, up to a sign, the product of the determinants of the $X_i$. We generalize this to arbitrary rings with appropriate"},"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":"math/9410207","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.RA","submitted_at":"1994-10-25T00:00:00Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"1542b7a68403a9ff4c50fba90e805d084c6b8ea52e596a7df4f85e64bd1fb085","abstract_canon_sha256":"d01276b3c943a2720ac9b5b4882c05788d6082f49ac8f6db288d14554858bf88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.002013Z","signature_b64":"vaqNAVc/HDDfuimC1nb04qj2V5LX1Nqfg6PIxPqipqhpL6Hs8B7Czm9omSkuDxFhvyujGC7DiyYIKMB8sPuSCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e9fe8089bde87961d3180f0dfe0bb2e08991d3975ae8d6fa24df29c601082ae","last_reissued_at":"2026-05-18T01:05:51.001508Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.001508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Matrix Vieta Theorem","license":"","headline":"","cross_cats":["math.OA"],"primary_cat":"math.RA","authors_text":"Albert Schwarz, Dmitry Fuchs","submitted_at":"1994-10-25T00:00:00Z","abstract_excerpt":"We consider generalizations of the Vieta formula (relating the coefficients of an algebraic equation to the roots) to the case of equations whose coefficients are order-$k$ matrices.\n  Specifically, we prove that if $X_1,\\dots ,X_n$ are solutions of an algebraic matrix equation $X^n+A_1X^{n-1}+\\dots +A_n=0,$ independent in the sense that they determine the coefficients $A_1,\\dots ,A_n$, then the trace of $A_1$ is the sum of the traces of the $X_i$, and the determinant of $A_n$ is, up to a sign, the product of the determinants of the $X_i$. We generalize this to arbitrary rings with appropriate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9410207","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/9410207","created_at":"2026-05-18T01:05:51.001590+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/9410207v1","created_at":"2026-05-18T01:05:51.001590+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9410207","created_at":"2026-05-18T01:05:51.001590+00:00"},{"alias_kind":"pith_short_12","alias_value":"B2P6QCE332DZ","created_at":"2026-05-18T12:25:47.102015+00:00"},{"alias_kind":"pith_short_16","alias_value":"B2P6QCE332DZMHJR","created_at":"2026-05-18T12:25:47.102015+00:00"},{"alias_kind":"pith_short_8","alias_value":"B2P6QCE3","created_at":"2026-05-18T12:25:47.102015+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/B2P6QCE332DZMHJRQDYN7YF3FY","json":"https://pith.science/pith/B2P6QCE332DZMHJRQDYN7YF3FY.json","graph_json":"https://pith.science/api/pith-number/B2P6QCE332DZMHJRQDYN7YF3FY/graph.json","events_json":"https://pith.science/api/pith-number/B2P6QCE332DZMHJRQDYN7YF3FY/events.json","paper":"https://pith.science/paper/B2P6QCE3"},"agent_actions":{"view_html":"https://pith.science/pith/B2P6QCE332DZMHJRQDYN7YF3FY","download_json":"https://pith.science/pith/B2P6QCE332DZMHJRQDYN7YF3FY.json","view_paper":"https://pith.science/paper/B2P6QCE3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/9410207&json=true","fetch_graph":"https://pith.science/api/pith-number/B2P6QCE332DZMHJRQDYN7YF3FY/graph.json","fetch_events":"https://pith.science/api/pith-number/B2P6QCE332DZMHJRQDYN7YF3FY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B2P6QCE332DZMHJRQDYN7YF3FY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B2P6QCE332DZMHJRQDYN7YF3FY/action/storage_attestation","attest_author":"https://pith.science/pith/B2P6QCE332DZMHJRQDYN7YF3FY/action/author_attestation","sign_citation":"https://pith.science/pith/B2P6QCE332DZMHJRQDYN7YF3FY/action/citation_signature","submit_replication":"https://pith.science/pith/B2P6QCE332DZMHJRQDYN7YF3FY/action/replication_record"}},"created_at":"2026-05-18T01:05:51.001590+00:00","updated_at":"2026-05-18T01:05:51.001590+00:00"}