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In this notation, a conjecture of Bl. Sendov claims that $r_n(\\beta) \\le 1$.\n  In this paper we investigate Sendov's conjecture near the unit circle, by computing constants $C_1$ and $C_2$ (depending only on $n$) such that $r_n(\\beta) \\sim 1 + C_1 (1-|\\beta|) + C_2 ("},"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/0310004","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CV","submitted_at":"2003-10-01T01:35:08Z","cross_cats_sorted":[],"title_canon_sha256":"4e9bd74bdbc75b82301105033753d0501bc24d6694c3d5012e93aa3081443387","abstract_canon_sha256":"2d3103c98512cf9ddefebe3ee37a48248a7cdb6e5dc0fee793b2f9f8d4309671"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:37:45.625274Z","signature_b64":"zx2MBzsAy/0pc6sJXidcESoL1HszvMjRdioGtXnAXik7QRZTkp73UR5XLZt0nzSEdA54DerskLFfI/s6u5nEAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"627ab5ac92307663121bcec9ac8a9bba17aec7cec1dda1d144bf5feb0c82e922","last_reissued_at":"2026-07-04T14:37:45.624883Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:37:45.624883Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A quadratic approximation to the Sendov radius near the unit circle","license":"","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Michael Miller","submitted_at":"2003-10-01T01:35:08Z","abstract_excerpt":"Define $S(n,\\beta)$ to be the set of complex polynomials of degree $n \\ge 2$ with all roots in the unit disk and at least one root at $\\beta$. 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Sendov claims that $r_n(\\beta) \\le 1$.\n  In this paper we investigate Sendov's conjecture near the unit circle, by computing constants $C_1$ and $C_2$ (depending only on $n$) such that $r_n(\\beta) \\sim 1 + C_1 (1-|\\beta|) + C_2 ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0310004","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/math/0310004/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":"math/0310004","created_at":"2026-07-04T14:37:45.624953+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0310004v1","created_at":"2026-07-04T14:37:45.624953+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0310004","created_at":"2026-07-04T14:37:45.624953+00:00"},{"alias_kind":"pith_short_12","alias_value":"MJ5LLLESGB3G","created_at":"2026-07-04T14:37:45.624953+00:00"},{"alias_kind":"pith_short_16","alias_value":"MJ5LLLESGB3GGEQ3","created_at":"2026-07-04T14:37:45.624953+00:00"},{"alias_kind":"pith_short_8","alias_value":"MJ5LLLES","created_at":"2026-07-04T14:37:45.624953+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/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI","json":"https://pith.science/pith/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI.json","graph_json":"https://pith.science/api/pith-number/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI/graph.json","events_json":"https://pith.science/api/pith-number/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI/events.json","paper":"https://pith.science/paper/MJ5LLLES"},"agent_actions":{"view_html":"https://pith.science/pith/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI","download_json":"https://pith.science/pith/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI.json","view_paper":"https://pith.science/paper/MJ5LLLES","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0310004&json=true","fetch_graph":"https://pith.science/api/pith-number/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI/graph.json","fetch_events":"https://pith.science/api/pith-number/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI/action/storage_attestation","attest_author":"https://pith.science/pith/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI/action/author_attestation","sign_citation":"https://pith.science/pith/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI/action/citation_signature","submit_replication":"https://pith.science/pith/MJ5LLLESGB3GGEQ3Z3E2ZCU3XI/action/replication_record"}},"created_at":"2026-07-04T14:37:45.624953+00:00","updated_at":"2026-07-04T14:37:45.624953+00:00"}