On tropical and Kapranov ranks of tropical matrices

From CRCG-Wiki

(Elena Rubei)

Let A be a tropical matrix (k+x) \times (k+x') for some k,x,x' natural numbers; let the tropical rank of A be k. We show that Kapranov rank is k too if x and x' are small; namely if we are in one of the following cases: a) k>=6 and x ,x'<=2 b) k =4,5, x<=2 and x'<=3 (or obviously the converse, i.e. x<=3 and x'<=2) c) k=3 and either x,x'<=3 or x<= 2, x'<=4 (or the converse). This answers, negatively, to Develin, Santos and Sturmfels' question whether there exists a matrix 5 \times 5 with tropical rank 3 and Kapranov rank strictly greater than 3.