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

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 $\varphi$.