The ultimate rank of tropical matrices

TL;DR

This paper introduces a new, well-defined notion of ultimate rank for tropical matrices that is independent of existing rank definitions, providing a polynomial-time computable formula and characterizations for semigroups.

Contribution

It defines the ultimate rank for tropical matrices, offers a simple polynomial-time formula, and characterizes semigroups with maximal ultimate rank.

Findings

Ultimate rank is independent of other rank notions.

A polynomial-time formula for ultimate rank is provided.

Algorithms to determine maximal ultimate rank in semigroups are developed.

Abstract

A tropical matrix is a matrix defined over the max-plus semiring. For such matrices, there exist several non-coinciding notions of rank: the row rank, the column rank, the Schein/Barvinok rank, the Kapranov rank, or the tropical rank, among others. In the present paper, we show that there exists a natural notion of ultimate rank for the powers of a tropical matrix, which does not depend on the underlying notion of rank. Furthermore, we provide a simple formula for the ultimate rank of a matrix which can therefore be computed in polynomial time. Then we turn our attention to finitely generated semigroups of matrices, for which our notion of ultimate rank is generalized naturally. We provide both combinatorial and geometric characterizations of semigroups having maximal ultimate rank. As a byproduct, we obtain a polynomial algorithm to decide if the ultimate rank of a finitely generated semigroup is maximal.