Canonical forms and moment-generating functions of plane polypols

TL;DR

This paper explores the relationship between canonical forms and moment-generating functions for plane domains bounded by rational algebraic arcs, extending known polygon dualities to curved polypols.

Contribution

It introduces a dual-geometric framework connecting canonical forms and Fantappie transforms for curved polypols, generalizing polygonal duality to nonlinear boundaries.

Findings

Normalized Fantappie transform of a polygon is the canonical form of its polar.

For curved polypols, the transform is a holonomic, branched period with controlled singularities.

Harmonic moment generating functions are one-dimensional restrictions of the Fantappiè transform.

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.