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Vandermonde Cells Through the Lens of Positive Geometry

Fatemeh Mohammadi, Sebastian Seemann

TL;DR

This work analyzes Vandermonde cells $\Pi_{n,d}$ through the lens of positive geometry, showing they share the cyclic-polytope combinatorics and can be treated within generalized positive-geometry frameworks. It provides explicit boundary parametrizations via Ursell patterns, derives algebraic boundary equations (with concrete planar cases), and develops planar canonical forms by subdividing into regions bounded by lines and cuspidal cubics, while placing Vandermonde cells inside Brown–Dúpont’s mixed Hodge theory to prove genus-zero boundaries and existence of canonical forms. In higher dimensions, the boundary structure becomes significantly more intricate, complicating direct canonical-form construction, though foundational birational properties (unirational boundaries, partition-counting) are established. The limiting object $\Pi_d$ reveals algebraic-to-analytic transition: its boundary is not semi-algebraic, motivating a dual-volume framework to define and approximate canonical forms via convex polytopes. Overall, the paper elevates Vandermonde cells as concrete, algebraic models within positive geometry, linking explicit boundary geometry, canonical forms, and Hodge-theoretic perspectives with implications for amplituhedron-like structures.

Abstract

We study the geometric and algebraic structure of Vandermonde cells, defined as images of the standard probability simplex under the Vandermonde map given by consecutive power sum polynomials. Motivated by their combinatorial equivalence to cyclic polytopes, which are well-known examples of positive geometries and tree amplituhedra, we investigate whether Vandermonde cells admit the structure of positive geometries. We derive explicit parametrizations and algebraic equations for their boundary components, extending known results from the planar case to arbitrary dimensions. By introducing a mild generalization of the notion of positive geometry, allowing singularities within boundary interiors, we show that planar Vandermonde cells naturally fit into this extended framework. Furthermore, we study Vandermonde cells in the setting of Brown-Dupont's mixed Hodge theory formulation of positive geometries, and show that they form a genus zero pair. These results provide a new algebraic and geometric understanding of Vandermonde cells, establishing them as promising examples within the emerging theory of positive geometries.

Vandermonde Cells Through the Lens of Positive Geometry

TL;DR

This work analyzes Vandermonde cells through the lens of positive geometry, showing they share the cyclic-polytope combinatorics and can be treated within generalized positive-geometry frameworks. It provides explicit boundary parametrizations via Ursell patterns, derives algebraic boundary equations (with concrete planar cases), and develops planar canonical forms by subdividing into regions bounded by lines and cuspidal cubics, while placing Vandermonde cells inside Brown–Dúpont’s mixed Hodge theory to prove genus-zero boundaries and existence of canonical forms. In higher dimensions, the boundary structure becomes significantly more intricate, complicating direct canonical-form construction, though foundational birational properties (unirational boundaries, partition-counting) are established. The limiting object reveals algebraic-to-analytic transition: its boundary is not semi-algebraic, motivating a dual-volume framework to define and approximate canonical forms via convex polytopes. Overall, the paper elevates Vandermonde cells as concrete, algebraic models within positive geometry, linking explicit boundary geometry, canonical forms, and Hodge-theoretic perspectives with implications for amplituhedron-like structures.

Abstract

We study the geometric and algebraic structure of Vandermonde cells, defined as images of the standard probability simplex under the Vandermonde map given by consecutive power sum polynomials. Motivated by their combinatorial equivalence to cyclic polytopes, which are well-known examples of positive geometries and tree amplituhedra, we investigate whether Vandermonde cells admit the structure of positive geometries. We derive explicit parametrizations and algebraic equations for their boundary components, extending known results from the planar case to arbitrary dimensions. By introducing a mild generalization of the notion of positive geometry, allowing singularities within boundary interiors, we show that planar Vandermonde cells naturally fit into this extended framework. Furthermore, we study Vandermonde cells in the setting of Brown-Dupont's mixed Hodge theory formulation of positive geometries, and show that they form a genus zero pair. These results provide a new algebraic and geometric understanding of Vandermonde cells, establishing them as promising examples within the emerging theory of positive geometries.

Paper Structure

This paper contains 17 sections, 20 theorems, 101 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Algebraic boundaries of Vandermonde cells
  3. Definitions and general boundary description
  4. Combinatorial and geometric properties
  5. Explicit equations and computational examples
  6. Planar Vandermonde cells
  7. Positive geometry and canonical forms
  8. Preliminaries: logarithmic forms, residues, and functoriality
  9. Positive geometry
  10. Canonical forms of planar Vandermonde cells via subdivision
  11. Higher-dimensional Vandermonde cells
  12. Positive geometries in the sense of Brown–Dúpont
  13. Hodge-theoretic setup
  14. Genus, combinatorial rank, and the iterated residue map
  15. Canonical form map and its properties
  16. ...and 2 more sections

Key Result

Theorem 2.1

The boundary $\,\mathrm{bd}\,\Pi_{n,d}$ of the Vandermonde cell is the image under the Vandermonde map $\nu_{n,d}$ of points of one of the following two types: Here each $m_i$ denotes the multiplicity of the corresponding value $x_i$, that is, the number of consecutive repetitions of $x_i$. Moreover, the preimages are unique up to permutation of coordinates.

Figures (2)

  • Figure 1: Semialgebraic sets arising in the subdivision of the planar Vandermonde cell from Example \ref{['Ex:GeometricArgumentforFirstCanForm']}.
  • Figure 2: The polytopal subdivision used to determine $\omega_{P_3}(x)$.

Theorems & Definitions (55)

  • Theorem 2.1: 1959_UrsellWonderfulGeometry
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4: WonderfulGeometry, Theorem 3.1
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Example 2.7
  • Proposition 2.8
  • ...and 45 more