{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:WUVMIV7RJJPTXLBABYEPB4YWP7","short_pith_number":"pith:WUVMIV7R","schema_version":"1.0","canonical_sha256":"b52ac457f14a5f3bac200e08f0f3167fc9311cb9a986e1cbfb8f5ca661cf26f0","source":{"kind":"arxiv","id":"2605.25992","version":1},"attestation_state":"computed","paper":{"title":"Root of a cubic polynomial as a power series in the discriminant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Jason Bland, Joel Rosenberg, Skip Garibaldi","submitted_at":"2026-05-25T16:11:43Z","abstract_excerpt":"An observation by J-P. Serre implies that cubic polynomials are unique among generic monic polynomials of degree 2 or higher in that they have a root that is a power series in the discriminant of the polynomial. We provide formulas for this root of a cubic that work in any characteristic. In the special case of a cubic real polynomial with positive discriminant, the series converges and therefore provides an explicit formula for a root; when that polynomial is depressed, the root we provide is the longest root. The proofs are a combination of elementary techniques from algebra, combinatorics, "},"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":"2605.25992","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2026-05-25T16:11:43Z","cross_cats_sorted":[],"title_canon_sha256":"c54eb349aff6a01f8b0ececf2a2cd04d99dbaae678916c4aacb16b47c4a9dee0","abstract_canon_sha256":"1ab8a9e88c01b9a533e7cbdc3877c97c151ef585db4a936cd12d5350fd6ce3e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:05:22.209991Z","signature_b64":"Z/Pgv1XwmoEEJBP3uY8lZOgqKl2FJiqGrOuuoYSb4zs6FLp4JnnnlR7JCge3R7QQEIJVJbuI392N/P8mXW5BCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b52ac457f14a5f3bac200e08f0f3167fc9311cb9a986e1cbfb8f5ca661cf26f0","last_reissued_at":"2026-05-26T02:05:22.209135Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:05:22.209135Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Root of a cubic polynomial as a power series in the discriminant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Jason Bland, Joel Rosenberg, Skip Garibaldi","submitted_at":"2026-05-25T16:11:43Z","abstract_excerpt":"An observation by J-P. Serre implies that cubic polynomials are unique among generic monic polynomials of degree 2 or higher in that they have a root that is a power series in the discriminant of the polynomial. We provide formulas for this root of a cubic that work in any characteristic. In the special case of a cubic real polynomial with positive discriminant, the series converges and therefore provides an explicit formula for a root; when that polynomial is depressed, the root we provide is the longest root. The proofs are a combination of elementary techniques from algebra, combinatorics, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25992","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.25992/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2605.25992","created_at":"2026-05-26T02:05:22.209276+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.25992v1","created_at":"2026-05-26T02:05:22.209276+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.25992","created_at":"2026-05-26T02:05:22.209276+00:00"},{"alias_kind":"pith_short_12","alias_value":"WUVMIV7RJJPT","created_at":"2026-05-26T02:05:22.209276+00:00"},{"alias_kind":"pith_short_16","alias_value":"WUVMIV7RJJPTXLBA","created_at":"2026-05-26T02:05:22.209276+00:00"},{"alias_kind":"pith_short_8","alias_value":"WUVMIV7R","created_at":"2026-05-26T02:05:22.209276+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/WUVMIV7RJJPTXLBABYEPB4YWP7","json":"https://pith.science/pith/WUVMIV7RJJPTXLBABYEPB4YWP7.json","graph_json":"https://pith.science/api/pith-number/WUVMIV7RJJPTXLBABYEPB4YWP7/graph.json","events_json":"https://pith.science/api/pith-number/WUVMIV7RJJPTXLBABYEPB4YWP7/events.json","paper":"https://pith.science/paper/WUVMIV7R"},"agent_actions":{"view_html":"https://pith.science/pith/WUVMIV7RJJPTXLBABYEPB4YWP7","download_json":"https://pith.science/pith/WUVMIV7RJJPTXLBABYEPB4YWP7.json","view_paper":"https://pith.science/paper/WUVMIV7R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.25992&json=true","fetch_graph":"https://pith.science/api/pith-number/WUVMIV7RJJPTXLBABYEPB4YWP7/graph.json","fetch_events":"https://pith.science/api/pith-number/WUVMIV7RJJPTXLBABYEPB4YWP7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WUVMIV7RJJPTXLBABYEPB4YWP7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WUVMIV7RJJPTXLBABYEPB4YWP7/action/storage_attestation","attest_author":"https://pith.science/pith/WUVMIV7RJJPTXLBABYEPB4YWP7/action/author_attestation","sign_citation":"https://pith.science/pith/WUVMIV7RJJPTXLBABYEPB4YWP7/action/citation_signature","submit_replication":"https://pith.science/pith/WUVMIV7RJJPTXLBABYEPB4YWP7/action/replication_record"}},"created_at":"2026-05-26T02:05:22.209276+00:00","updated_at":"2026-05-26T02:05:22.209276+00:00"}