{"paper":{"title":"Law of large numbers for the discriminant of random polynomials","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Marcus Michelen, Oren Yakir","submitted_at":"2025-06-13T20:21:51Z","abstract_excerpt":"Let $f_n$ be a random polynomial of degree $n$, whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let $\\Delta(f_n)$ denote the discriminant of $f_n$, that is $\\Delta(f_n) = A^{2n-2}\\prod_{i < j} (\\alpha_j - \\alpha_i)^2$ where $A$ is the leading coefficient of $f_n$ and $\\alpha_1,\\ldots\\alpha_n$ are its roots. We prove that with high probability $$|\\Delta(f_n)| = n^{2n} e^{-{\\sf D}_\\ast n(1+o(1))}$$ as $n\\to \\infty$, for some explicit universal constant ${\\sf D}_\\ast>0$. A key step in the proof is an analytic representation for the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.12206","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/2506.12206/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"}