Equality of tropical rank and dimension for tropical linear series

TL;DR

The paper resolves a central question in tropical linear series by proving the exact alignment between tropical rank and topological dimension: for any subsemimodule $M\subseteq R(D)$, $r_{\mathrm{trop}}(M)=\dim(M)$ and $r_{\mathrm{trop}}|(D,M)|=\dim|(D,M)|$, tying combinatorial rank to geometric size. The authors establish this through a reduction to finitely generated subsemimodules, certificates of tropical independence derived from nonlinear fixed-point operators, and finite evaluation maps, culminating in a polyhedral description of the linear system $|(D,M)|$ and a criterion that divisorial and tropical ranks coincide exactly when the system is pure dimensional. They also connect these results to a broader framework, discuss algorithmic implications (including hardness results for computing tropical rank), and pose questions about higher-dimensional generalizations and the computability of tropical independence. Overall, the work clarifies the structure of tropical linear series on metric graphs and lays groundwork for further combinatorial, topological, and computational investigations in tropical geometry.

Abstract

The tropical rank of a semimodule of rational functions on a metric graph mirrors the concept of rank in linear algebra. Defined in terms of the maximal number of tropically independent elements within the semimodule, this quantity has remained elusive due to the challenges of computing it in practice. In this note, we establish that the tropical rank is, in fact, precisely equal to the topological dimension of the semimodule, one more than the dimension of the associated linear system of divisors. Moreover, we show that the equality of divisorial and tropical ranks in the definition of tropical linear series is equivalent to the pure dimensionality of the corresponding linear system. We conclude with several complementary results and questions on combinatorial, topological, and computability properties of the tropical rank.

Figures (2)

• Figure 1: Construction of infinitely many functions using three different slopes.
• Figure 2: A parametric family of functions $f \in R(D)$ (left). Fragment of the cross section $\mleft\{(f(x_1), f(x_3)) \bigm| f \in M, f(x_2) = 0\mright\}$ with an infinite upper Pareto set depicted in red (right).

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof