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Positive Polytopes with Few Facets in the Grassmannian

Dmitrii Pavlov, Kristian Ranestad

TL;DR

This work investigates adjoint hypersurfaces for positive polytopes in the Grassmannian $Gr(2,4)$ obtained by intersecting the positive Grassmannian $Gr_{\ge 0}(2,4)$ with one or two additional half-spaces, introducing the notions of positive pentahedra and hexahedra. The authors develop a residue-based framework to study adjoint polynomials and canonical forms, proving that a unique adjoint exists for pentahedra under mild genericity conditions, and establishing bounds (up to six dimensions) on the family of adjoints for hexahera, with evidence suggesting tighter limits in practice. They introduce a refined combinatorial classification for hexaherda to account for interactions with the Plücker quadric, and provide several detailed examples showing how positive geometries can arise even when adjoints are not unique. The results illuminate the boundary structure and canonical forms of these Grassmannian-intersected polytopes and raise open questions about topology, computation, and broader generalizations to higher Grassmannians. Overall, the paper deepens the connection between residual arrangements, adjoint interpolation, and positive geometries in a Grassmannian setting, with explicit constructions and computer-assisted verifications guiding the theory.

Abstract

In this article we study adjoint hypersurfaces of geometric objects obtained by intersecting simple polytopes with few facets in $\mathbb{P}^5$ with the Grassmannian $\mathrm{Gr}(2,4)$. These generalize the positive Grassmannian, which is the intersection of $\mathrm{Gr}(2,4)$ with the simplex. We show that if the resulting object has five facets, it is a positive geometry and the adjoint hypersurface is unique. For the case of six facets we show that the adjoint hypersurface is not necessarily unique and give an upper bound on the dimension of the family of adjoints. We illustrate our results with a range of examples. In particular, we show that even if the adjoint is not unique, a positive hexahedron can still be a positive geometry.

Positive Polytopes with Few Facets in the Grassmannian

TL;DR

This work investigates adjoint hypersurfaces for positive polytopes in the Grassmannian obtained by intersecting the positive Grassmannian with one or two additional half-spaces, introducing the notions of positive pentahedra and hexahedra. The authors develop a residue-based framework to study adjoint polynomials and canonical forms, proving that a unique adjoint exists for pentahedra under mild genericity conditions, and establishing bounds (up to six dimensions) on the family of adjoints for hexahera, with evidence suggesting tighter limits in practice. They introduce a refined combinatorial classification for hexaherda to account for interactions with the Plücker quadric, and provide several detailed examples showing how positive geometries can arise even when adjoints are not unique. The results illuminate the boundary structure and canonical forms of these Grassmannian-intersected polytopes and raise open questions about topology, computation, and broader generalizations to higher Grassmannians. Overall, the paper deepens the connection between residual arrangements, adjoint interpolation, and positive geometries in a Grassmannian setting, with explicit constructions and computer-assisted verifications guiding the theory.

Abstract

In this article we study adjoint hypersurfaces of geometric objects obtained by intersecting simple polytopes with few facets in with the Grassmannian . These generalize the positive Grassmannian, which is the intersection of with the simplex. We show that if the resulting object has five facets, it is a positive geometry and the adjoint hypersurface is unique. For the case of six facets we show that the adjoint hypersurface is not necessarily unique and give an upper bound on the dimension of the family of adjoints. We illustrate our results with a range of examples. In particular, we show that even if the adjoint is not unique, a positive hexahedron can still be a positive geometry.

Paper Structure

This paper contains 12 sections, 5 theorems, 40 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Positive polytopes in $\mathrm{Gr}(2,4)$
  4. Combinatorial classification of positive hexahedra
  5. Adjoints
  6. Pentahedra
  7. Hexahedra
  8. Examples
  9. A positive geometry.
  10. A positive geometry.
  11. A positive geometry.
  12. Open questions and future directions

Key Result

Theorem 4.2

Suppose $c_I\neq 0$ for all $I$, and suppose the hyperplane $H$ defined by $h(\mathbf{p}) = 0$ intersects every facet of $\mathrm{Gr}_{\geq 0}(2,4)$. Then there exists a unique adjoint of the positive pentahedron $S = P \cap \mathrm{Gr}(2,4)$, and $S$ is a positive geometry. The numerator of its can

Figures (4)

  • Figure 1: Residual arrangement (red) and adjoint curve (orange) of a pentagon in the plane.
  • Figure 2: Intersections between the components of the residual arrangement in Example \ref{['ex:54']}
  • Figure 3: Intersections between the components of the residual arrangement in Example \ref{['ex:schub']}
  • Figure 4: Intersections between the components of the residual arrangement in Example \ref{['ex:manyadj']}

Theorems & Definitions (21)

  • Definition 2.1: Residual arrangement of a polytope
  • Definition 2.2: Adjoint hypersurface/polynomial of a polytope
  • Definition 2.3: Positive geometries
  • Definition 2.4: Positive Grassmannian
  • Definition 3.1: Positive polytopes
  • Remark 3.2
  • Definition 3.3: Residual arrangement of a positive polytope
  • Definition 4.1: Adjoints of positive polytopes in $\mathrm{Gr}(2,4)$
  • Theorem 4.2
  • proof
  • ...and 11 more