On bases of tropical Pl\"ucker functions

We consider functions $f:B\to \text{Rset}$ that obey tropical analogs of classical Pl"ucker relations on minors of a matrix. The most general set $B$ that we deal with in this paper is of the form ${x\in \Zset^n\colon 0\le x\le a, m\le x_1+…+x_n\le m’}$ (a rectangular integer box ``truncated from below and above’'). We construct a basis for the set $\text{Tscr}$ of tropical Pl"ucker functions on $B$, a subset $\text{Bscr}\subseteq B$ such that the restriction map $\text{Tscr}\to {\text{Rset}}^{\text{Bscr}}$ is bijective. Also we characterize, in terms of the restriction to the basis, the classes of submodular, so-called skew-submodular, and discrete concave functions in $\text{Tscr}$, discuss a tropical analogue of the Laurentness property, and present other results.