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Root polytopes, tropical types, and toric edge ideals

Ayah Almousa, Anton Dochtermann, Ben Smith

TL;DR

This work builds a bridge between tropical geometry and commutative algebra by studying arrangements of tropical hyperplanes in the tropical torus and their associated type/cotype data. It develops a robust dictionary between tropical complexes (and their bounded subcomplexes) and subdivisions of root polytopes $Q_B$ and generalized permutohedra $P_B$, using cellular/cocellular resolutions to extract algebraic invariants. A key contribution is the Alexander duality linking fine cotype ideals to initial ideals of lattice (toric edge) ideals, enabling new dimension and regularity bounds for toric edge rings of bipartite graphs and providing new proofs of known results. The paper also gives concrete recipes for volumes, syzygies, and linear resolutions in the bipartite setting, with extensions to graphic tropical hyperplane arrangements and G-parking function ideals. Overall, the results offer a unifying tropical framework for understanding volumes, syzygies, and homological properties of toric and monomial ideals arising from root polytopes and generalized permutohedra.

Abstract

We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data, analogous to the covectors of an oriented matroid. By work of Develin-Sturmfels and Fink-Rincón, these `tropical complexes' are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig-Sanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among `type ideals' which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a corresponding initial ideal of the lattice ideal of the underlying root polytope. This leads to novel ways of studying algebraic properties of various monomial and toric ideals, as well as relating them to combinatorial and geometric properties. In particular, our methods of studying the dimension of the tropical complex leads to new formulas for homological invariants of toric edge ideals of bipartite graphs, which have been extensively studied in the commutative algebra community.

Root polytopes, tropical types, and toric edge ideals

TL;DR

This work builds a bridge between tropical geometry and commutative algebra by studying arrangements of tropical hyperplanes in the tropical torus and their associated type/cotype data. It develops a robust dictionary between tropical complexes (and their bounded subcomplexes) and subdivisions of root polytopes and generalized permutohedra , using cellular/cocellular resolutions to extract algebraic invariants. A key contribution is the Alexander duality linking fine cotype ideals to initial ideals of lattice (toric edge) ideals, enabling new dimension and regularity bounds for toric edge rings of bipartite graphs and providing new proofs of known results. The paper also gives concrete recipes for volumes, syzygies, and linear resolutions in the bipartite setting, with extensions to graphic tropical hyperplane arrangements and G-parking function ideals. Overall, the results offer a unifying tropical framework for understanding volumes, syzygies, and homological properties of toric and monomial ideals arising from root polytopes and generalized permutohedra.

Abstract

We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data, analogous to the covectors of an oriented matroid. By work of Develin-Sturmfels and Fink-Rincón, these `tropical complexes' are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig-Sanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among `type ideals' which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a corresponding initial ideal of the lattice ideal of the underlying root polytope. This leads to novel ways of studying algebraic properties of various monomial and toric ideals, as well as relating them to combinatorial and geometric properties. In particular, our methods of studying the dimension of the tropical complex leads to new formulas for homological invariants of toric edge ideals of bipartite graphs, which have been extensively studied in the commutative algebra community.
Paper Structure (37 sections, 48 theorems, 83 equations, 11 figures)

This paper contains 37 sections, 48 theorems, 83 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Generalizations to root polytopes
  3. Our contributions
  4. Resolutions of type ideals
  5. Toric edge rings
  6. Running example
  7. Organization and notation
  8. Notation
  9. Tropical hyperplanes, root polytopes, and generalized permutohedra
  10. Tropical Hyperplane Arrangements
  11. Types and cotypes
  12. Subdivisions of root polytopes and generalized permutohedra
  13. The bounded complex and topology of types
  14. The recession cone and dimension of the bounded complex
  15. Topology of types
  16. ...and 22 more sections

Key Result

Theorem A

Let $B \subset K_{d,n}$ be a connected bipartite graph. Then for any $d \times n$ tropical matrix $A$ satisfying $B = B_A$ we have Furthermore, if $A$ is sufficiently generic then we have equality in the above expression.

Figures (11)

  • Figure 1: The bipartite graph $B_A$, hyperplane arrangement ${\mathcal{H}} = \mathcal{H}(A)$, and bounded complex $\mathcal{B}(A)$ associated to the tropical matrix $A$ from Example \ref{['ex: running']}. The points in $\mathbb{R}^{3}/\mathbb{R}\mathbb{1}$ are labelled by their representative with $p_1 = 0$.
  • Figure 2: The generalized permutohedron $P_B$ along with the mixed subdivision dual to the hyperplane arrangement in Figure \ref{['fig: tropHAex']}. The monomial ideal generated by the vertices is minimally supported by this complex.
  • Figure 3: The tropical complex and bounded complex corresponding to the tropical hyperplane arrangement in Figure \ref{['fig: tropHAex']}.
  • Figure 4: The tropical types corresponding to the tropical complex and distinguished cells in Figure \ref{['fig: tropicalcomplex']}.
  • Figure 5: The tropical hyperplane arrangements $\mathcal{H}(A)$ and $\mathcal{H}(A^T)$. Their bounded complexes, shown in bold, are isomorphic.
  • ...and 6 more figures

Theorems & Definitions (121)

  • Theorem A: Proposition \ref{['prop: bounded+complex+dimension']}, Corollary \ref{['cor: bounded+complex+equality']}
  • Theorem B: Propositions \ref{['prop: typesLabelCells']}, \ref{['prop: cocellularRes']}, \ref{['prop: cotypeCellRes']}
  • Theorem C: Proposition \ref{['prop: cotypeIdealAD']}
  • Theorem D: Theorem \ref{['thm: tropicalCplxRegularity']}, Corollary \ref{['cor: krulldim']}
  • Theorem E: Theorem \ref{['thm: linear']}
  • Example 1.1
  • Definition 2.1: Tropical Semiring
  • Definition 2.2: Tropical Hyperplane Arrangement
  • Definition 2.3: Support of a Hyperplane
  • Definition 2.4: Bipartite graphs from tropical matrices
  • ...and 111 more