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

Abstract

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

16 pages. arXiv admin note: text overlap with arXiv:2110.08091