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 [Formula: see text] by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves [Formula: see text], the rational function semifield of [Formula: see text] is finitely generated as a [Formula: see text]-algebra, where [Formula: see text] stands for the pull-back of the rational function semifield of [Formula: see text] by [Formula: see text].