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arxiv: 2605.10864 · v2 · pith:ZBBQIZ7Xnew · submitted 2026-05-11 · 🧮 math.AG · math.CO· math.CV

Canonical forms and moment-generating functions of plane polypols

Boris Shapiro This is my paper

Pith reviewed 2026-05-20 22:41 UTC · model grok-4.3

classification 🧮 math.AG math.COmath.CV
keywords canonical formsFantappie transformpolypolspositive geometryholonomic periodsmoment-generating functionsalgebraic arcspolarity
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0 comments X

The pith

For domains bounded by algebraic arcs, the Fantappie transform is a holonomic period controlled by vertices and dual curves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines canonical forms and moment-generating functions for plane domains with boundaries made of rational algebraic arcs. It shows that the polarity relation linking these two objects, which holds for polygons, extends to curved polypols. In the curved case the transform is no longer a simple rational function but a holonomic branched period. This matters because it provides a uniform way to handle positive geometries and their associated transforms beyond straight-edged cases. Explicit examples like sectors and half-disks illustrate how the singularities arise from vertex hyperplanes and projective duals of the curved parts.

Core 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.

What carries the argument

The dual-geometric mechanism of polarity that identifies the normalized Fantappie transform of a polypol with its canonical form, extended to cases with nonlinear boundary arcs.

Load-bearing premise

The dual-geometric mechanism of polarity continues to operate even when some boundary components are nonlinear algebraic arcs.

What would settle it

Compute the Fantappie transform explicitly for a half-disk and check whether its singularities match the expected vertex hyperplanes and the dual curve of the circular arc.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies canonical forms in positive geometry and normalized moment-generating functions (Fantappié transforms) for plane polypols bounded by rational algebraic arcs. For polygons the normalized Fantappié transform equals the canonical form of the polar polygon by polarity. For polypols with genuinely curved algebraic boundary components the same dual-geometric polarity mechanism persists, but the transform is realized as a holonomic, generally branched period whose singularities are located at vertex hyperplanes and the projective dual curves of the nonlinear arcs. Explicit constructions are given for sectors and half-disks; one-dimensional restrictions of the same transform yield harmonic moment-generating functions.

Significance. If the explicit computations hold, the work supplies a concrete extension of the polarity relation between canonical forms and Fantappié transforms from rational polygons to polypols whose boundaries include nonlinear algebraic arcs. By exhibiting the transform as a holonomic period controlled by vertex hyperplanes and projective duals, the paper links positive geometry with the theory of holonomic functions and their singularity loci. The self-contained examples on sectors and half-disks furnish verifiable test cases that demonstrate survival of the dual mechanism without requiring rationality or global single-valuedness of the transform. This supplies a useful bridge between canonical forms, moment problems, and algebraic geometry.

minor comments (3)
  1. The abstract contains the typographical string 'Fantappi`e'; this should be corrected to the standard spelling 'Fantappié' throughout the manuscript.
  2. In the half-disk example, the notation for the projective dual curve and its relation to the period integral could be made more explicit so that the location of the singularities is immediately visible to readers.
  3. A brief remark on the precise order of the holonomic differential equation satisfied by the Fantappié transform in the curved case would help readers connect the construction to the general theory of holonomic D-modules.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. We are pleased that the connections to holonomic periods and the explicit examples for sectors and half-disks are viewed as useful bridges between positive geometry and algebraic analysis.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit examples

full rationale

The paper's central claim—that the dual-geometric polarity mechanism extends to polypols with nonlinear algebraic arcs, yielding a holonomic Fantappié transform controlled by vertex hyperplanes and projective dual curves—is established through direct, explicit computations on concrete cases such as sectors and half-disks. These constructions identify the transform with a branched period without any reduction to fitted parameters, self-definitions, or load-bearing self-citations. The one-dimensional harmonic moment-generating functions follow directly from the same explicit restrictions. The derivation relies on established notions of polarity and Fantappié transforms but does not presuppose the target result; the examples are independent and falsifiable by direct calculation, rendering the chain non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no free parameters, axioms, or invented entities are identifiable from the given text. The work appears to rely on standard background notions from positive geometry and the theory of Fantappie transforms without introducing new ad-hoc objects.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    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.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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    Burman, R

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