Local theory of functions on tropical curves in $\mathbb R^n$

We first develop the local theory of functions on $\mathbb R^n$ defined by tropical Laurent polynomials. We study the structure of the semiring of functions, where two functions are identified when they coincide on a neighborhood of a fixed point. We see that this semiring is closely related to the semiring of functions defined by Boolean Laurent polynomials. Then we develop the local theory of functions on tropical curves. We construct a contravariant functor from the category of 1-dimensional tropical fans to the category of certain homomorphisms of semirings. As an application, we discuss about the smoothness of 1-dimensional tropical fans at the origin.