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arXiv · 2605.10864

Canonical forms and moment-generating functions of plane polypols

Boris Shapiro
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Abstract

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.

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Boris Shapiro. 2026-05-11. Canonical forms and moment-generating functions of plane polypols. https://arxiv.org/abs/2605.10864

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