Linear independence over tropical semirings and beyond
Commutative Algebra
2009-12-13 v1 Combinatorics
Abstract
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.
Cite
@article{arxiv.0812.3496,
title = {Linear independence over tropical semirings and beyond},
author = {Marianne Akian and Stephane Gaubert and Alexander Guterman},
journal= {arXiv preprint arXiv:0812.3496},
year = {2009}
}