{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:VYIZT27NZKJ4GZ4A6RPWMIOCV6","short_pith_number":"pith:VYIZT27N","schema_version":"1.0","canonical_sha256":"ae1199ebedca93c36780f45f6621c2af8e38367bfcb27f147c5c07c928b251a4","source":{"kind":"arxiv","id":"1202.2475","version":2},"attestation_state":"computed","paper":{"title":"On the speed of convergence of Newton's method for complex polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Dierk Schleicher, Magnus Aspenberg, Todor Bilarev","submitted_at":"2012-02-11T21:02:36Z","abstract_excerpt":"We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\\mathcal{S}_d$ of $3.33d\\log^2 d(1 + o(1))$ points with the following universal property: for every normalized polynomial of degree $d$ there are $d$ starting points in $\\mathcal{S}_d$ whose Newton iterations find all the roots with a low number of iterations: if the roots are uniformly and independently distributed, we show that with probability at least $1-2/d$ the number of iterations for these $d$ starting poi"},"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":"1202.2475","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-11T21:02:36Z","cross_cats_sorted":[],"title_canon_sha256":"35e5c9aa564918090e94080abc98e8d94db4bafaeef4ed1adc17268e0d11aee6","abstract_canon_sha256":"25766cf057c494724f98f4bb5799ad0a90578dfd9eb43033c009847df22acff0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:57.800223Z","signature_b64":"7P6rYs9cA8tsRd+fPyVDGIhVeA6dxhkVV/CWA/mjS8GC/iSU0gW0ucPB9Jlm1BDmzt6/PV+M/DOu5dNOklXqBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae1199ebedca93c36780f45f6621c2af8e38367bfcb27f147c5c07c928b251a4","last_reissued_at":"2026-05-18T01:18:57.799820Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:57.799820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the speed of convergence of Newton's method for complex polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Dierk Schleicher, Magnus Aspenberg, Todor Bilarev","submitted_at":"2012-02-11T21:02:36Z","abstract_excerpt":"We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\\mathcal{S}_d$ of $3.33d\\log^2 d(1 + o(1))$ points with the following universal property: for every normalized polynomial of degree $d$ there are $d$ starting points in $\\mathcal{S}_d$ whose Newton iterations find all the roots with a low number of iterations: if the roots are uniformly and independently distributed, we show that with probability at least $1-2/d$ the number of iterations for these $d$ starting poi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2475","kind":"arxiv","version":2},"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":"1202.2475","created_at":"2026-05-18T01:18:57.799887+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.2475v2","created_at":"2026-05-18T01:18:57.799887+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.2475","created_at":"2026-05-18T01:18:57.799887+00:00"},{"alias_kind":"pith_short_12","alias_value":"VYIZT27NZKJ4","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"VYIZT27NZKJ4GZ4A","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"VYIZT27N","created_at":"2026-05-18T12:27:25.539911+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/VYIZT27NZKJ4GZ4A6RPWMIOCV6","json":"https://pith.science/pith/VYIZT27NZKJ4GZ4A6RPWMIOCV6.json","graph_json":"https://pith.science/api/pith-number/VYIZT27NZKJ4GZ4A6RPWMIOCV6/graph.json","events_json":"https://pith.science/api/pith-number/VYIZT27NZKJ4GZ4A6RPWMIOCV6/events.json","paper":"https://pith.science/paper/VYIZT27N"},"agent_actions":{"view_html":"https://pith.science/pith/VYIZT27NZKJ4GZ4A6RPWMIOCV6","download_json":"https://pith.science/pith/VYIZT27NZKJ4GZ4A6RPWMIOCV6.json","view_paper":"https://pith.science/paper/VYIZT27N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.2475&json=true","fetch_graph":"https://pith.science/api/pith-number/VYIZT27NZKJ4GZ4A6RPWMIOCV6/graph.json","fetch_events":"https://pith.science/api/pith-number/VYIZT27NZKJ4GZ4A6RPWMIOCV6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VYIZT27NZKJ4GZ4A6RPWMIOCV6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VYIZT27NZKJ4GZ4A6RPWMIOCV6/action/storage_attestation","attest_author":"https://pith.science/pith/VYIZT27NZKJ4GZ4A6RPWMIOCV6/action/author_attestation","sign_citation":"https://pith.science/pith/VYIZT27NZKJ4GZ4A6RPWMIOCV6/action/citation_signature","submit_replication":"https://pith.science/pith/VYIZT27NZKJ4GZ4A6RPWMIOCV6/action/replication_record"}},"created_at":"2026-05-18T01:18:57.799887+00:00","updated_at":"2026-05-18T01:18:57.799887+00:00"}