Tropical totally positive matrices

We start by presenting Gaubert's symmetrized tropical semiring, which defines a tropical additive-inverse and uses it to resolve tropical singularity. Then, we recall properties of totally positive matrices over rings, define tropical total positivity and total non-negativity of matrices using the symmetrized structure, and state combinatorial and algebraic properties of these matrices. By studying the tropical semiring via valuation on the field of Puiseux series, we relate the tropical properties to the classical ones.
Joint work with Stephane Gaubert