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Then the distribution of the complex roots of $G_n$ concentrates in probability, as $n\\to\\infty$, to two centered circles and is uniform in the argument as $n\\to\\infty$. The radii of the circles are $|\\xi_0/\\xi_\\tau|^{1/\\tau}$ and $|\\xi_\\tau/\\xi_n|^{1/(n-\\tau)}$, where $\\xi_\\tau$ denotes the coefficient with the maximum modulus."},"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":"1104.5360","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-04-28T11:37:25Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"dd08d134a561319072fd759450599c2b155c0c189fe027e97a4d9ff0ce21fef8","abstract_canon_sha256":"2d16d4fa7c61b3d947f1beb40af5839c623a85c3e11f63c74b8a98d8f4cd955e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:21.628674Z","signature_b64":"RjzSxkbqAnRbYBwYeOZ9uKp6HKGit/IxulbxjUF3rqy4Um2f8X17pyNgiAlVxNgil728aaHMY+6ifK51stF3Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38802bad323096472cc24ba698459ed3a2333c53d29c3d1811ed80aff6892ad7","last_reissued_at":"2026-05-18T04:23:21.628178Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:21.628178Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, Friedrich G\\\"otze","submitted_at":"2011-04-28T11:37:25Z","abstract_excerpt":"Consider a random polynomial $G_n(z)=\\xi_nz^n+...+\\xi_1z+\\xi_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\\log(1+\\log(1+|\\xi_0|))$ has a slowly varying tail. Then the distribution of the complex roots of $G_n$ concentrates in probability, as $n\\to\\infty$, to two centered circles and is uniform in the argument as $n\\to\\infty$. The radii of the circles are $|\\xi_0/\\xi_\\tau|^{1/\\tau}$ and $|\\xi_\\tau/\\xi_n|^{1/(n-\\tau)}$, where $\\xi_\\tau$ denotes the coefficient with the maximum modulus."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.5360","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":"1104.5360","created_at":"2026-05-18T04:23:21.628247+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.5360v1","created_at":"2026-05-18T04:23:21.628247+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.5360","created_at":"2026-05-18T04:23:21.628247+00:00"},{"alias_kind":"pith_short_12","alias_value":"HCACXLJSGCLE","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"HCACXLJSGCLEOLGC","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"HCACXLJS","created_at":"2026-05-18T12:26:30.835961+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/HCACXLJSGCLEOLGCJOTJQRM62O","json":"https://pith.science/pith/HCACXLJSGCLEOLGCJOTJQRM62O.json","graph_json":"https://pith.science/api/pith-number/HCACXLJSGCLEOLGCJOTJQRM62O/graph.json","events_json":"https://pith.science/api/pith-number/HCACXLJSGCLEOLGCJOTJQRM62O/events.json","paper":"https://pith.science/paper/HCACXLJS"},"agent_actions":{"view_html":"https://pith.science/pith/HCACXLJSGCLEOLGCJOTJQRM62O","download_json":"https://pith.science/pith/HCACXLJSGCLEOLGCJOTJQRM62O.json","view_paper":"https://pith.science/paper/HCACXLJS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.5360&json=true","fetch_graph":"https://pith.science/api/pith-number/HCACXLJSGCLEOLGCJOTJQRM62O/graph.json","fetch_events":"https://pith.science/api/pith-number/HCACXLJSGCLEOLGCJOTJQRM62O/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HCACXLJSGCLEOLGCJOTJQRM62O/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HCACXLJSGCLEOLGCJOTJQRM62O/action/storage_attestation","attest_author":"https://pith.science/pith/HCACXLJSGCLEOLGCJOTJQRM62O/action/author_attestation","sign_citation":"https://pith.science/pith/HCACXLJSGCLEOLGCJOTJQRM62O/action/citation_signature","submit_replication":"https://pith.science/pith/HCACXLJSGCLEOLGCJOTJQRM62O/action/replication_record"}},"created_at":"2026-05-18T04:23:21.628247+00:00","updated_at":"2026-05-18T04:23:21.628247+00:00"}