{"paper":{"title":"A Separation Method for Quartic Positivity and the Valid Region of Gram-Charlier densities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A separation method supplies necessary and sufficient conditions for a quartic polynomial to stay positive everywhere, which then yields simpler analytic expressions for the valid region of Gram-Charlier densities.","cross_cats":[],"primary_cat":"cs.SC","authors_text":"ByoungSeon Choi, Jung Chan Lee, Taehun Kim","submitted_at":"2026-02-12T04:48:27Z","abstract_excerpt":"The positivity of the Gram-Charlier probability density function has been a subject of extensive study for decades. Since Barton and Dennis (1952) introduced numerical positivity conditions, no analytic closed-form expression was available until Kwon (2019, 2022) proposed analytic solutions for the valid region of Gram-Charlier densities. Despite the significance of the analytical solutions, the expressions remain algebraically complex. As these conditions for the Gram-Charlier densities are determined by a quartic polynomial, it is essential to investigate its positivity. In this work, necess"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"necessary and sufficient conditions for the positivity of a quartic polynomial are derived through a separation method. Based on these conditions, more concise analytic expressions for the positivity of the Gram-Charlier density are proposed.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The separation method fully captures all cases of positivity for the specific quartic polynomials that arise from the Gram-Charlier expansion, without missing boundary cases or requiring additional restrictions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A separation method for quartic positivity produces more concise analytic conditions for the valid region of Gram-Charlier densities.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A separation method supplies necessary and sufficient conditions for a quartic polynomial to stay positive everywhere, which then yields simpler analytic expressions for the valid region of Gram-Charlier densities.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ae0583fd61bb77b7077cee21005adf72f03aae0b546bce6ae81bc579b579a0d5"},"source":{"id":"2603.00073","kind":"arxiv","version":2},"verdict":{"id":"3e91eccc-b5a4-4dd6-8680-db48dbcb4fa6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T02:12:07.394893Z","strongest_claim":"necessary and sufficient conditions for the positivity of a quartic polynomial are derived through a separation method. Based on these conditions, more concise analytic expressions for the positivity of the Gram-Charlier density are proposed.","one_line_summary":"A separation method for quartic positivity produces more concise analytic conditions for the valid region of Gram-Charlier densities.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The separation method fully captures all cases of positivity for the specific quartic polynomials that arise from the Gram-Charlier expansion, without missing boundary cases or requiring additional restrictions.","pith_extraction_headline":"A separation method supplies necessary and sufficient conditions for a quartic polynomial to stay positive everywhere, which then yields simpler analytic expressions for the valid region of Gram-Charlier densities."},"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"}