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Bulletin of the London Mathematical Society· 2026Q1

A tropical version of Martens' theorem for metric graphs

G. Capobianco, Angelina Zheng

Short summary

A new class of counterexamples, Martens-special trees of cycles, is constructed for a tropical version of Martens' theorem, but the theorem is proven to hold for metric graphs with a stricter degree assumption in the Brill-Noether rank.

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Abstract

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.

The authors' abstract, as published at the source. Bulletin of the London Mathematical Society, 2026 · DOI ↗

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Field: Computational Theory and Mathematics

Computational Theory and MathematicsComputer Science