{"paper":{"title":"Discriminants of derivatives and symmetric difference polynomials","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Boris Shapiro","submitted_at":"2026-05-25T11:56:11Z","abstract_excerpt":"Let $P$ be a monic polynomial of degree $n$ with roots $x_1,\\ldots,x_n$. We study the discriminants of the derivatives $P^{(k)}$ as symmetric translation-invariant polynomials in the original roots. The general ``square-graph cone'' positivity problem was formulated by Alexandersson and Shapiro. The main result of this note proves this conjecture for the terminal cubic family $k=n-3$: we give an explicit positive square-graph expansion for $\\disc(P^{(n-3)})$. We also record closed central-moment formulas for the terminal quadratic, cubic and quartic cases, introduce normalized terminal polynom"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25743","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/2605.25743/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"}