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We prove a conjecture of Pemantle and Rivin [arXiv:1109.5975] that the empirical measure $\\mu_n:=\\frac 1{n-1}\\sum_{P_n'(z)=0} \\delta_z$ counting the complex zeros of the derivative $P_n'$ converges in probability to $\\mu$, as $n\\to\\infty$."},"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":"1206.6692","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-06-28T13:51:15Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"4ad57b86dfe257436558282f5390688340f8010f6c3952b3bbb77ddae004c2f7","abstract_canon_sha256":"5bc23114aff819bb2a615d959fa85277edcc7d8528892a0afb798da3354386ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:23.898669Z","signature_b64":"fjbveIFYwwN8V/fhyjW7Y3SZlvLwNQNVApkMFey2fpnXzoxn98uctsT48BZZ+JL9v13cOGIZwGJI6duDuJpMDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"998b9a078796be6fd6d26ef8f0723af56a1efe2ac653a76f1a9376c22c4c66e9","last_reissued_at":"2026-05-18T03:44:23.898068Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:23.898068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical points of random polynomials with independent identically distributed roots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.PR","authors_text":"Zakhar Kabluchko","submitted_at":"2012-06-28T13:51:15Z","abstract_excerpt":"Let $X_1,X_2,...$ be independent identically distributed random variables with values in $\\C$. Denote by $\\mu$ the probability distribution of $X_1$. Consider a random polynomial $P_n(z)=(z-X_1)...(z-X_n)$. We prove a conjecture of Pemantle and Rivin [arXiv:1109.5975] that the empirical measure $\\mu_n:=\\frac 1{n-1}\\sum_{P_n'(z)=0} \\delta_z$ counting the complex zeros of the derivative $P_n'$ converges in probability to $\\mu$, as $n\\to\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6692","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":"1206.6692","created_at":"2026-05-18T03:44:23.898151+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.6692v2","created_at":"2026-05-18T03:44:23.898151+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.6692","created_at":"2026-05-18T03:44:23.898151+00:00"},{"alias_kind":"pith_short_12","alias_value":"TGFZUB4HS27G","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"TGFZUB4HS27G7VWS","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"TGFZUB4H","created_at":"2026-05-18T12:27:23.164592+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/TGFZUB4HS27G7VWSN34PA4R26V","json":"https://pith.science/pith/TGFZUB4HS27G7VWSN34PA4R26V.json","graph_json":"https://pith.science/api/pith-number/TGFZUB4HS27G7VWSN34PA4R26V/graph.json","events_json":"https://pith.science/api/pith-number/TGFZUB4HS27G7VWSN34PA4R26V/events.json","paper":"https://pith.science/paper/TGFZUB4H"},"agent_actions":{"view_html":"https://pith.science/pith/TGFZUB4HS27G7VWSN34PA4R26V","download_json":"https://pith.science/pith/TGFZUB4HS27G7VWSN34PA4R26V.json","view_paper":"https://pith.science/paper/TGFZUB4H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.6692&json=true","fetch_graph":"https://pith.science/api/pith-number/TGFZUB4HS27G7VWSN34PA4R26V/graph.json","fetch_events":"https://pith.science/api/pith-number/TGFZUB4HS27G7VWSN34PA4R26V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TGFZUB4HS27G7VWSN34PA4R26V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TGFZUB4HS27G7VWSN34PA4R26V/action/storage_attestation","attest_author":"https://pith.science/pith/TGFZUB4HS27G7VWSN34PA4R26V/action/author_attestation","sign_citation":"https://pith.science/pith/TGFZUB4HS27G7VWSN34PA4R26V/action/citation_signature","submit_replication":"https://pith.science/pith/TGFZUB4HS27G7VWSN34PA4R26V/action/replication_record"}},"created_at":"2026-05-18T03:44:23.898151+00:00","updated_at":"2026-05-18T03:44:23.898151+00:00"}