Semiring isomorphisms between rational function semifields of tropical curves induce isomorphisms between tropical curves

TL;DR

This paper establishes a deep connection between semiring isomorphisms of rational function semifields and the geometric structure of tropical curves, showing how algebraic symmetries reflect curve isomorphisms.

Contribution

It proves that semiring isomorphisms induce expansive maps between tropical curves and characterizes automorphism groups of these curves in algebraic terms.

Findings

Semiring isomorphisms induce expansive maps respecting zeros and poles.

Automorphism groups of tropical curves correspond to algebra automorphisms of their semifields.

Complete description of all semiring automorphisms for rational function semifields.

Abstract

We prove that a semiring isomorphism between the rational function semifields of two tropical curves induces an expansive map between those tropical curves. This semiring isomorphism and the expansive map respect zeros and poles of rational functions with their degrees. As a corollary, we show that the automorphism group of a tropical curve is isomorphic to the $\boldsymbol{T}$-algebra automorphism group of its rational function semifield, where $\boldsymbol{T} := (\boldsymbol{R} \cup \{ -\infty \}, \operatorname{max}, +)$ is the tropical semifield. Finally, we describe all semiring automorphisms of rational function semifields of all tropical curves.