Three notions of tropical rank for symmetric matrices

TL;DR

This paper introduces three new concepts of tropical rank for symmetric matrices, analyzing their properties and relationships with tropical secant sets, including dimensions and convex hulls, within tropical geometry.

Contribution

It defines and studies three notions of tropical rank for symmetric matrices, linking them to tropical secant sets and tropical varieties like the Grassmannian.

Findings

Determined the dimension of each tropical secant set.

Analyzed the convex hulls of the tropical varieties.

Identified the smallest secant set equal to the convex hull in most cases.

Abstract

We introduce and study three different notions of tropical rank for symmetric and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull.