When do the r-by-r minors of a matrix form a tropical basis?

TL;DR

This paper characterizes when the r-by-r minors of a matrix form a tropical basis, providing a complete answer to a previously posed question in tropical geometry.

Contribution

It establishes precise conditions under which minors form a tropical basis, resolving an open problem in the field.

Findings

Minors form a tropical basis if r<4 or r=min(d,n).

Minors form a tropical basis if r=4 and min(d,n)<7.

The result fully answers the question posed by Chan, Jensen, and Rubei.

Abstract

We show that the r-by-r minors of a d-by-n matrix of variables form a tropical basis of the ideal they generate if and only if r<4, or r=min(d,n), or else r=4 and min(d,n)<7. This answers a question asked by M. Chan, A. Jensen, and E. Rubei.