{"paper":{"title":"On the geometry of random lemniscates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.AT","math.MG","math.PR"],"primary_cat":"math.CV","authors_text":"Antonio Lerario, Erik Lundberg","submitted_at":"2016-01-11T01:29:43Z","abstract_excerpt":"We investigate the geometry of a random rational lemniscate $\\Gamma$, the level set $\\{|r(z)|=1\\}$ on the Riemann sphere of the modulus of a random rational function $r$. We assign a probability distribution to the space of rational functions $r=p/q$ of degree $n$ by sampling $p$ and $q$ independently from the complex Kostlan ensemble of random polynomials of degree $n$.\n  We prove that the average \\emph{spherical length} of $\\Gamma$ is $\\frac{\\pi^2}{2} \\sqrt{n},$ which is proportional to the square root of the maximal spherical length. We also provide an asymptotic for the average number of p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02295","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":""},"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"}