Galois actions for semifield extensions and Galois coverings on tropical curves

TL;DR

This paper explores Galois actions on semifield extensions and establishes a correspondence between Galois coverings of tropical curves and Galois actions on their rational function semifields, linking algebraic and tropical geometry.

Contribution

It characterizes Galois coverings of tropical curves via Galois actions on their rational function semifields, connecting tropical morphisms with semifield extension theory.

Findings

Galois actions on semifield extensions are characterized by invariance properties.

A finite harmonic morphism is Galois if and only if the induced action on rational functions is Galois.

The paper establishes a criterion linking tropical curve coverings with semifield Galois extensions.

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 $\varphi$.