On bases of tropical Plücker functions
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We consider functions $f:B\to\Rset$ that obey tropical analogs of classical Plücker 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 $\Tscr$ of tropical Plücker functions on $B$, a subset $\Bscr\subseteq B$ such that the restriction map $\Tscr\to\Rset^\Bscr$ 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 $\Tscr$, discuss a tropical analogue of the Laurentness property, and present other results.
44 pages. This is a revision of the original version, where some improvements are done and new results are added (in particular, the classes of submodular and discrete concave tropical Plücker functions are characterized, and a tropical analogue of the Laurent phenomenon is shown
44 pages. This is a revision of the original version, where some improvements are done and new results are added (in particular, the classes of submodular and discrete concave tropical Plücker functions are characterized, and a tropical analogue of the Laurent phenomenon is shown