On the Kapranov ranks of tropical matrices
The author investigates the Kapranov rank functions of tropical matrices for different ground fields. For any infinite ground field he shows that the rank-product inequality holds for the Kapranov rank, and he proves that the Kapranov rank respects Green's preorders on the semigroup of tropical \(n\)-by-\(n\) matrices. The rank-product inequality is shown to fail for the Kapranov rank over any finite ground field. He provides an example of a 7-by-7 0-1-matrix whose Kapranov rank is independent of the ground field, equals 6, and exceeds tropical rank.
scientific article; zbMATH DE number 6034453