The image of a tropical linear space

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.

19 pages, 1 figure