Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Tropical Variety of Symmetric Rank 2 Matrices

May Cai, Kisun Lee, Josephine Yu

TL;DR

This work provides a complete combinatorial and geometric description of the tropicalization of the variety of symmetric rank $2$ matrices. It introduces symbic trees as the indexing objects for the maximal cones of a simplicial fan and proves shellability of the resulting complex, yielding topological clarity. It also delivers explicit generating functions for enumerating regular symbic trees and develops a Cayley-embedding framework to describe the associated algebraic matroid, including conjectures that may simplify the basis structure. Collectively, the results deepen understanding of symmetric tropical rank, with implications for symmetric matrix completion and tropical geometry.

Abstract

We study the tropicalization of the variety of symmetric rank two matrices. Analogously to the result of Markwig and Yu for general tropical rank two matrices, we show that it has a simplicial complex structure as the space of symmetric bicolored trees and that this simplicial complex is shellable. We also discuss some matroid structures arising from this space and present generating functions for the number of symmetric bicolored trees.

The Tropical Variety of Symmetric Rank 2 Matrices

TL;DR

This work provides a complete combinatorial and geometric description of the tropicalization of the variety of symmetric rank matrices. It introduces symbic trees as the indexing objects for the maximal cones of a simplicial fan and proves shellability of the resulting complex, yielding topological clarity. It also delivers explicit generating functions for enumerating regular symbic trees and develops a Cayley-embedding framework to describe the associated algebraic matroid, including conjectures that may simplify the basis structure. Collectively, the results deepen understanding of symmetric tropical rank, with implications for symmetric matrix completion and tropical geometry.

Abstract

We study the tropicalization of the variety of symmetric rank two matrices. Analogously to the result of Markwig and Yu for general tropical rank two matrices, we show that it has a simplicial complex structure as the space of symmetric bicolored trees and that this simplicial complex is shellable. We also discuss some matroid structures arising from this space and present generating functions for the number of symmetric bicolored trees.
Paper Structure (6 sections, 14 theorems, 19 equations, 8 figures)

This paper contains 6 sections, 14 theorems, 19 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Symmetric bicolored trees
  3. Simplicial fan structure
  4. Shellability
  5. Symbic tree enumeration
  6. The algebraic matroid of rank 2 symmetric matrices

Key Result

Theorem 2.4

Let $A$ be a symmetric tropical matrix of symmetric tropical rank $2$. Then the associated bicolored tree $T_A$ is a symbic tree. Conversely, every symbic tree arises from a symmetric tropical matrix of symmetric tropical rank $2$, and the matrix is unique up to simultaneous tropical scaling of rows

Figures (8)

  • Figure 1: The embedding of the matrices in Example \ref{['ex:identity']}. The left is the embedding of the $3 \times 3$ identity matrix. All the internal edges are fixed points of the involution from Definition \ref{['def:symbic']} on the left tree, while on the right the single fixed point internal edge is thicker.
  • Figure 2: The space of $3 \times 3$ symmetric tropical rank $2$ matrices, where $a, b \ge 0$. There are 9 top dimensional cells in the tropical hypersurface of the $3 \times 3$ symmetric determinant. The symbic trees give a finer polyhedral structure with $12$ top dimensional cells.
  • Figure 3: A $4 + 4$ symbic tree. A chosen fixed vertex $O$ can be used to parameterize the set of matrices corresponding to this tree. See Example \ref{['ex:4x4matrix']}.
  • Figure 4: The trees (a) and (b) are symbic trees, and the trees (c) and (d) are the result of removing $4$ from (a) and (b) respectively. The tree (c) is symbic, while (d) is not as the bicoloring condition is not satisfied. Thus (a) is not brittle, but (b) has a brittle twig$(1,2)$.
  • Figure 5: This picture shows that only caterpillar branches are needed for the algebraic matroid. See Proposition \ref{['prop:caterpillar-basis']}. Here $A$, $B$, and $C$ are bicolored subtrees, not leaves.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Example 2.2
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • proof : Proof of Theorem \ref{['thm:symbic']}
  • Lemma 3.1
  • proof
  • ...and 24 more