半域扩张的Galois作用和热带曲线上的Galois覆盖
Galois actions for semifield extensions and Galois coverings on tropical curves
摘要：
对于一个半域扩张$T /S$，有限群$G$在$T$上的作用是伽罗瓦，如果$(1)$的$G$不变子半域是$S$，并且不变半域重合的$G$的$(2)$子群相等.我们证明了：对于热带曲线$φ: \varGamma \to \varGamma^{\prime}$和有限群$G$在$\varGamma$上的等距作用之间的有限调和态射，$φ$是$G$ -Galois当且仅当$G$在$\varGamma$的有理函数半域$\operatorname{Rat}(\varGamma)$上的自然作用对于半域扩张$\operatorname{Rat}(\varGamma) / φ^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$是Galois，其中，$φ^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$代表$φ$对$\operatorname{Rat}(\varGamma^{\prime})$的回撤。
英文摘要：
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 $φ: \varGamma \to \varGamma^{\prime}$ and an isometric action of a finite group $G$ on $\varGamma$, $φ$ 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 $\operatorname{Rat}(\varGamma) / φ^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$, where $φ^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ stands for the pull-back of $\operatorname{Rat}(\varGamma^{\prime})$ by $φ$.
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