Rank-determining sets of metric graphs

TL;DR

This paper introduces the concept of rank-determining sets in metric graphs, proves their existence, and provides criteria and algorithms for identifying such sets, advancing the understanding of divisor theory in tropical geometry.

Contribution

It defines rank-determining sets for metric graphs, proves their existence constructively, and develops criteria and algorithms for their identification.

Findings

Finite rank-determining sets exist for metric graphs.

A criterion for identifying rank-determining sets is formulated.

An algorithm to derive $v_0$-reduced divisors is developed.

Abstract

A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph $\Gamma$ is an element of the free abelian group on $\Gamma$. The rank of a divisor on a metric graph is a concept appearing in the Riemann-Roch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and Zharkov. We define a \emph{rank-determining set} of a metric graph $\Gamma$ to be a subset $A$ of $\Gamma$ such that the rank of a divisor $D$ on $\Gamma$ is always equal to the rank of $D$ restricted on $A$. We show constructively in this paper that there exist finite rank-determining sets. In addition, we investigate the properties of rank-determining sets in general and formulate a criterion for rank-determining sets. Our analysis is a based on an algorithm to derive the $v_0$-reduced divisor from any effective divisor in the same linear system.