Galois actions for semifield extensions and Galois coverings on tropical curves

Abstract

For a semifield extension $T /S$, an action of a finite group $G$ on $T$ is Galois if $(1)$ the $G$-invariant subsemifield of $T$ is $S$ and $(2)$ subgroups of $G$ whose invariant semifields coincide are equal. We show that for a finite harmonic morphism between tropical curves $\varphi : \varGamma \to \varGamma^{\prime}$ and an isometric action of a finite group $G$ on $\varGamma$, $\varphi$ is $G$-Galois if and only if the natural action of $G$ on the rational function semifield $\operatorname{Rat}(\varGamma)$ of $\varGamma$ induced by the action of $G$ on $\varGamma$ is Galois for the semifield extension $\operatorname{Rat}(\varGamma) / \varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$, where $\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ stands for the pull-back of $\operatorname{Rat}(\varGamma^{\prime})$ by $\varphi$.