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Let $S(n)$ be the largest possible shortest length of a path in $E_f$ joining $0$ to $\\partial\\mathbb{D}$, where the maximum is taken over all such polynomials of degree $n$. We prove that, for all sufficiently large $n$, $c\\sqrt{\\log n}\\leq S(n)\\leq \\pi n$ with an absolute constant $c>0$. This proves the qualitative unboundedness predicted by Erd\\H{o}s. The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analy"},"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":"2606.19178","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2026-06-17T15:19:13Z","cross_cats_sorted":[],"title_canon_sha256":"0f84f198c37a7b50f7e4a6129a080a470dfbd98ebe716ecf567494630a2035fd","abstract_canon_sha256":"e6e929ed904dfd1448173259b9a902114e5f30187b82ee9d34a8d65777fd3fe9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:12:07.192146Z","signature_b64":"P9/p9WMdNtIzAupssyPa/zkCtbBhh2b9rxfoAyJGPBBxtzJLSKUvX+HNvEHkMCO1lScRX9qaJoqTFhsoVtDyDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8e236ccb6902df2d02c541a2563cd682a32b9425ea02eb73dc72ec96d96def5","last_reissued_at":"2026-06-19T16:12:07.191796Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:12:07.191796Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shortest paths in polynomial lemniscate sublevel sets and a problem of Erd\\H{o}s","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Venkata Siddharth Pendyala","submitted_at":"2026-06-17T15:19:13Z","abstract_excerpt":"Let $f(z)=\\prod_{j=1}^{n}(z-a_j)$ be monic, with all zeros in the closed unit disk, and put $E_f=\\{z\\in\\mathbb{C}: |z|\\leq 1,\\ |f(z)|\\leq 1\\}$. Let $S(n)$ be the largest possible shortest length of a path in $E_f$ joining $0$ to $\\partial\\mathbb{D}$, where the maximum is taken over all such polynomials of degree $n$. We prove that, for all sufficiently large $n$, $c\\sqrt{\\log n}\\leq S(n)\\leq \\pi n$ with an absolute constant $c>0$. This proves the qualitative unboundedness predicted by Erd\\H{o}s. 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