Rank of divisors on tropical curves

TL;DR

This paper explores the properties of divisors on tropical curves, providing a combinatorial proof of the Riemann-Roch theorem, confirming a conjecture relating graph and metric graph divisor ranks, and developing an algorithm for rank computation.

Contribution

It offers a new combinatorial proof of the Riemann-Roch theorem for tropical curves and confirms Baker's conjecture on divisor ranks, along with an algorithm for computing these ranks.

Findings

Elementary proof of Riemann-Roch theorem for tropical curves

Confirmation of Baker's conjecture on divisor ranks

Development of an algorithm for computing divisor ranks

Abstract

We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, an elementary proof of the Riemann-Roch theorem for tropical curves, similar to the recent proof of the Riemann-Roch theorem for graphs by Baker and Norine, is presented. In addition, a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph is confirmed, and an algorithm for computing the rank of a divisor on a tropical curve is constructed.