Rational function semifields of tropical curves are finitely generated over the tropical semifield

TL;DR

This paper proves that the rational function semifield of a tropical curve is finitely generated over the tropical semifield, and extends this to finite harmonic morphisms between tropical curves, providing explicit generators.

Contribution

It establishes finite generation of rational function semifields for tropical curves and their behavior under finite harmonic morphisms, with explicit generators provided.

Findings

Rational function semifield of a tropical curve is finitely generated.

Finite harmonic morphisms induce finitely generated structures.

Explicit finite generating sets are constructed.

Abstract

We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield $\boldsymbol{T} := ( \boldsymbol{R} \cup \{ - \infty \}, \operatorname{max}, +)$ by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves $\varphi : \varGamma \to \varGamma^{\prime}$, the rational function semifield of $\varGamma$ is finitely generated as a $\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$-algebra, where $\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ stands for the pull-back of the rational function semifield of $\varGamma^{\prime}$ by $\varphi$.