A tropical version of Martens' theorem for metric graphs

TL;DR

This paper investigates a tropical analogue of Martens' theorem for metric graphs, providing counterexamples and conditions under which the conjecture holds, advancing understanding in tropical Brill--Noether theory.

Contribution

It introduces new classes of graphs serving as counterexamples and establishes conditions where the tropical Martens' conjecture is valid.

Findings

Counterexamples to the conjecture via Martens-special trees of cycles

The conjecture holds under stricter degree assumptions in Brill--Noether rank

Generalization of previous counterexamples to broader classes of graphs

Abstract

We study the conjecture stated by Jensen and Len on a tropical version on Martens' theorem via the Brill--Noether rank of a tropical curve. We recall Coppens' counterexample of Martens-special chain of cycles, and we generalize the construction defining another class of graphs, Martens-special trees of cycles, for which the conjecture does not hold in a similar setting. These are not the only counterexamples. However, we prove that the conjecture holds for all metric graphs with a stricter assumption on the degree in the Brill--Noether rank.