Rank functions of tropical matrices

Abstract

We consider rank functions important in tropical linear algebra: the tropical rank, equal to the topological dimension of the tropical linear span of the columns of a given matrix, the factor rank, equal to the smallest number of vectors containing these columns in their span, the Kapranov rank, related to problems of tropical algebraic geometry, and the determinantal and Gondran–Minoux ranks, which generalize the classical linear algebraic notion of rank. We are interested in studying the arithmetic properties of the rank functions, their mutual behavior, and the related computational problems. We discuss different open problems related to rank functions of tropical matrices and illustrate them by the number of examples.