The image of a tropical linear space

TL;DR

This paper introduces the concept of a tropical image of a tropical linear space under a matrix, providing a new way to understand their structure and applications like constructing stable sums.

Contribution

It defines the tropical image of a tropical linear space, proves its realizability under certain conditions, and applies it to construct stable sums without disjoint bases.

Findings

The tropical image contains the usual image under mild conditions.

The tropical image is realizable if the original space is realizable.

Application to stable sums of tropical linear spaces.

Abstract

Given a tropical linear space $L \subseteq \mathbb{T}^n$ and a matrix $A \in \mathbb{T}^{m \times n}$, the image $AL$ of $L$ under $A$ is typically not a tropical linear space. We introduce a tropical linear space $\mathrm{tropim}_A(L)$, the tropical image, containing $AL$. We show under mild hypotheses that $\mathrm{tropim}_A(L)$ is realizable if $L$ is and apply the tropical image to construct the stable sum of two tropical linear spaces without a disjoint pair of bases.