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 $\phi :\Gamma \to {\Gamma }^{\prime }$ and an isometric action of a finite group $G$ on $\Gamma$, $\phi$ is $G$-Galois if and only if the natural action of $G$ on the rational function semifield $\mathrm{Rat}(\Gamma )$ of $\Gamma$ induced by the action of $G$ on $\Gamma$ is Galois for the semifield extension $\mathrm{Rat}(\Gamma )/{\phi }^{\ast }(\mathrm{Rat}({\Gamma }^{\prime }))$, where ${\phi }^{\ast }(\mathrm{Rat}({\Gamma }^{\prime }))$ stands for the pull-back of $\mathrm{Rat}({\Gamma }^{\prime })$ by $\phi$.

12 pages, 1 figure. arXiv admin note: text overlap with arXiv:2201.06603