{"paper":{"title":"Sharp order in Erd\\H{o}s's minimum-area problem for polynomial lemniscates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Venkata Siddharth Pendyala","submitted_at":"2026-06-13T20:46:04Z","abstract_excerpt":"For a monic polynomial $p$, its filled unit lemniscate is the planar set ${z: |p(z)|<1}$. Let $\\kappa_n(K,1)$ denote the least possible area of this set among monic polynomials of degree $n$ whose zeros lie in a compact set $K$. We prove that there are absolute constants $c,C>0$ such that $c/\\log n \\leq \\kappa_n(\\overline{\\mathbb{D}},1) \\leq \\kappa_n(\\mathbb{T},1) \\leq C/\\log n$. Thus the recently established lower bound has the correct order, even when all zeros are required to lie on the unit circle. The upper bound is obtained by combining a quantitative Faber-polynomial separator for a thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17097","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/2606.17097/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"}