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

摘要

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 one of the following conditions holds: (i) r ≤ 3, (ii) r = min{. d, n} ≤ 6. This answers a question raised by M. Chan, A. Jensen, and E. Rubei.