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In the asymptotic regime when $n\\to\\infty$, with high probability there is a critical point of $Q_n$ which is very close to $\\xi_0$. We localize the position of this critical point by proving that the difference between $\\xi_0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(\\xi_0))$ and variance of or"},"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":"1807.02140","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-07-05T18:34:09Z","cross_cats_sorted":[],"title_canon_sha256":"f04b45229af55e55448321386d4c88e28097c57862d8756b05e178e0f5a3a3b4","abstract_canon_sha256":"4ee31036d2b977aa3088cab5c6e82cd771621c94d53dffd8db03c229c8259877"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:22.999830Z","signature_b64":"qiACMb4ND6Mk/mJqUu5vlPEqZ3KEJY3CgZG04E6TJikjaC4mTW7oSWas4dr8t2m6u/Wb5L9T97oAluHv/CrMDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"44474e4a31b73276202fe5dca63233249c87e9d0d0c241c0d56d750453f68c47","last_reissued_at":"2026-05-18T00:11:22.999278Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:22.999278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distances between zeroes and critical points for random polynomials with i.i.d. zeroes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hauke Seidel, Zakhar Kabluchko","submitted_at":"2018-07-05T18:34:09Z","abstract_excerpt":"Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\\xi_0,\\xi_1,\\ldots,\\xi_n$ in the complex plane. 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