{"paper":{"title":"The lemniscate tree of a random polynomial","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.CV"],"primary_cat":"math.PR","authors_text":"Boris Hanin, Erik Lundberg, Michael Epstein","submitted_at":"2018-06-01T19:45:21Z","abstract_excerpt":"To each generic complex polynomial $p(z)$ there is associated a labeled binary tree (here referred to as a \"lemniscate tree\") that encodes the topological type of the graph of $|p(z)|$. The branching structure of the lemniscate tree is determined by the configuration (i.e., arrangement in the plane) of the singular components of those level sets $|p(z)|=t$ passing through a critical point.\n  In this paper, we address the question \"How many branches appear in a typical lemniscate tree?\" We answer this question first for a lemniscate tree sampled uniformly from the combinatorial class and second"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00521","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"}