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We prove that a semiring isomorphism between the rational function semifieldsof two tropical curves induces an expansive map between those tropical curves.This semiring isomorphism and the expansive map respect zeros and poles ofrational functions with their degrees. As a corollary, we show that theautomorphism 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 allsemiring automorphisms of rational function semifields of all tropical curves.

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