Canonical forms and moment-generating functions of plane polypols
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918508883869696 |
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| author | Shapiro, Boris |
| author_facet | Shapiro, Boris |
| contents | 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 vertex hyperplanes and by the projective dual curves of the nonlinear boundary components. We give explicit examples, including sectors and half-disks, and explain how harmonic moment generating functions arise as one-dimensional restrictions of the same Fantappi`e transform. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_10864 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Canonical forms and moment-generating functions of plane polypols Shapiro, Boris Algebraic Geometry Combinatorics Complex Variables Primary 14P10, 52B11, Secondary 14Q05, 30E05, 65D17 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 vertex hyperplanes and by the projective dual curves of the nonlinear boundary components. We give explicit examples, including sectors and half-disks, and explain how harmonic moment generating functions arise as one-dimensional restrictions of the same Fantappi`e transform. |
| title | Canonical forms and moment-generating functions of plane polypols |
| topic | Algebraic Geometry Combinatorics Complex Variables Primary 14P10, 52B11, Secondary 14Q05, 30E05, 65D17 |
| url | https://arxiv.org/abs/2605.10864 |