{"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. The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.19178","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/2606.19178/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"}