{"paper":{"title":"Canonical forms and moment-generating functions of plane polypols","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The polarity relation between canonical forms and Fantappie transforms for polygons extends to curved polypols, where the transform becomes a holonomic branched period controlled by vertex hyperplanes and dual curves.","cross_cats":["math.CO","math.CV"],"primary_cat":"math.AG","authors_text":"Boris Shapiro","submitted_at":"2026-05-11T17:12:34Z","abstract_excerpt":"We study two closely related objects associated with plane domains bounded by rational algebraic arcs: canonical forms in the sense of positive geometry and normalized moment-generating functions, or Fantappie transforms. For polygons these objects are related by polarity: the normalized Fantappie transform of a polygon is the canonical form of the polar polygon. For genuinely curved polypols the same dual-geometric mechanism survives, but the transform is no longer a rational logarithmic canonical form; rather, it is a holonomic, generally branched period whose singularities are controlled by"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For genuinely curved polypols the same dual-geometric mechanism survives, but the transform is no longer a rational logarithmic canonical form; rather, it is a holonomic, generally branched period whose singularities are controlled by vertex hyperplanes and by the projective dual curves of the nonlinear boundary components.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The dual-geometric polarity mechanism that works for polygons continues to govern the relation between canonical forms and Fantappie transforms when the boundary arcs are genuinely curved rational curves.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For plane polypols the normalized Fantappie transform is a holonomic generally branched period whose singularities are controlled by vertex hyperplanes and projective dual curves of the nonlinear boundary components.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The polarity relation between canonical forms and Fantappie transforms for polygons extends to curved polypols, where the transform becomes a holonomic branched period controlled by vertex hyperplanes and dual curves.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7c9d509dd1d626de0ecc1db458d04a8392914238b57b40fbfc75aeb2674a0842"},"source":{"id":"2605.10864","kind":"arxiv","version":2},"verdict":{"id":"ce9951fc-ea42-4b9c-af2f-ee7ecf8b6885","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T03:56:15.133194Z","strongest_claim":"For genuinely curved polypols the same dual-geometric mechanism survives, but the transform is no longer a rational logarithmic canonical form; rather, it is a holonomic, generally branched period whose singularities are controlled by vertex hyperplanes and by the projective dual curves of the nonlinear boundary components.","one_line_summary":"For plane polypols the normalized Fantappie transform is a holonomic generally branched period whose singularities are controlled by vertex hyperplanes and projective dual curves of the nonlinear boundary components.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The dual-geometric polarity mechanism that works for polygons continues to govern the relation between canonical forms and Fantappie transforms when the boundary arcs are genuinely curved rational curves.","pith_extraction_headline":"The polarity relation between canonical forms and Fantappie transforms for polygons extends to curved polypols, where the transform becomes a holonomic branched period controlled by vertex hyperplanes and dual curves."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.10864/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T14:34:08.586294Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T10:31:17.448830Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T08:56:19.918687Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"8d2897d48329910e95ca7205c803b2a059cad99a4e883f28355c8a5bfdf28f89"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"1f86b6520b3ac557470aaafbcfee90f609dd12437fe6590610559008024714af"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}