%0 Journal Article
%T Linear independence over tropical semirings and beyond
%A Marianne Akian
%A Stephane Gaubert
%A Alexander Guterman
%J Mathematics
%D 2008
%I arXiv
%X We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.
%U http://arxiv.org/abs/0812.3496v1