Idempotent Analysis, Tropical Convexity and Reduced Divisors

Abstract

We investigate a canonical extension of a conventional combinatorial notion of reduced divisors to a notion of tropical projections, which can be defined as the unique minimizers of the so-called $B$-pseudonorms with respect to compact tropical convex sets. In this paper, we build the foundation of a theory of idempotent analysis using tropical projections and obtain a series of subsequent results, e.g. tropical retracts, construction of compact tropical convex sets and a set-theoretical characterization of tropical weak independence. In particular, we prove a tropical version of Mazur's Theorem on closed tropical convex hulls and discover a fixed point theorem for tropical projections. As the main application of our machinery of tropical convexity analysis, we investigate the divisor theory on metric graphs based on tropical projections. We extend the notion of linear systems and redefine the notion of reduced divisors to all linear systems instead of only to complete linear systems. Moreover, we explore the correspondence between reduced divisor maps to dominant tropical trees and harmonic morphisms to metric trees. Furthermore, we propose a notion called the geometric rank for linear systems on metric graphs which resolves the discrepancy between the interpretations of gonality of metric graphs using the conventional Baker-Norine rank function and using harmonic morphisms to metric trees.