Tropical totally positive matrices (Q1794085)

A real matrix is called totally positive (resp., totally nonnegative) if all its minors are positive (resp., nonnegative). This paper investigates tropical analogues of totally positive and totally nonnegative matrices, by considering the images by the valuation of the corresponding classes of matrices over a non-Archimedean field. The classical theory of positive and non-negative matrices can be found in [\textit{S. M. Fallat} and \textit{C. R. Johnson}, Totally nonnegative matrices. Princeton, NJ: Princeton University Press (2011; Zbl 1390.15001); \textit{S. Fomin} and \textit{A. Zelevinsky}, Math. Intell. 22, No. 1, 23--33 (2000; Zbl 1052.15500)]. One of the main results of the article provides a correspondence between tropical totally positive matrices with the Monge matrices, defined by the positivity of \(2\times 2\) tropical minors. Monge matrices arise in optimal transportation problems. A characterization of Monge matrices can be found in [\textit{R. E. Burkard} et al., Discrete Appl. Math. 70, No. 2, 95--161 (1996; Zbl 0856.90091); \textit{M. Fiedler}, Linear Algebra Appl. 413, No. 1, 177--188 (2006; Zbl 1090.15016)]. The article comprises several results and applications of the study of totally positive matrices; in particular, relations between tropical total positivity and planar networks.