Rank-determining sets of metric graphs

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 Γ is an element of the free abelian group on Γ. 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 rank-determining set of a metric graph Γ to be a subset A of Γ such that the rank of a divisor D on Γ 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 based on an algorithm to derive the v 0 -reduced divisor from any effective divisor in the same linear system.