{"paper":{"title":"The arc length of a random lemniscate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CV"],"primary_cat":"math.PR","authors_text":"Erik Lundberg, Koushik Ramachandran","submitted_at":"2016-10-31T05:43:26Z","abstract_excerpt":"A polynomial lemniscate is a curve in the complex plane defined by $\\{z \\in \\mathbb{C}:|p(z)|=t\\}$. Erd\\\"os, Herzog, and Piranian posed the extremal problem of determining the maximum length of a lemniscate $\\Lambda=\\{ z \\in \\mathbb{C}:|p(z)|=1\\}$ when $p$ is a monic polynomial of degree $n$. In this paper, we study the length and topology of a random lemniscate whose defining polynomial has independent Gaussian coefficients. In the special case of the Kac ensemble we show that the length approaches a nonzero constant as $n \\rightarrow \\infty$. We also show that the average number of connected"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.09791","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"}