4 1. TROPICS x1 0 Figure 1. 0 ⊕ (0 x1) In order to write these formulas in a more invariant way, in what follows we shall often make use of the notation M = Zn, MR = M ⊗Z R, N = HomZ(M, Z), NR = N ⊗Z R. We denote evaluation of n ∈ N on m ∈ M by n, m . We shall often use the notion of index of an element m ∈ M \ {0} this is the largest positive integer r such that there exists m ∈ M with rm = m. If the index of m is 1, we say m is primitive. With this notation, we can view a tropical function as a map f : MR → R written, for S ⊆ N a finite set, as f(z) = n∈S anzn := min{an + n, z| n ∈ S}. Now V (f) will be a union of codimension one polyhedra in Rn. Here, by a polyhedron, we mean: Definition 1.1. A polyhedron σ in MR is a finite intersection of closed half- spaces. A face of a polyhedron is a subset given by the intersection of σ with a hyperplane H such that σ is contained in a half-space with boundary H. The boundary ∂σ of σ is the union of all proper faces of σ, and the interior Int(σ) of σ is σ \ ∂σ. The polyhedron σ is a lattice polyhedron if it is an intersection of half-spaces defined over Q and all vertices of σ lie in M. A polytope is a compact polyhedron. Returning to V (f), each codimension one polyhedron making up V (f) separates two domains of linearity of f, in one of which f is given by a monomial with exponent n ∈ N and in the other by a monomial with exponent n ∈ N. Then the weight of this polyhedron in V (f) is the index of n − n. We then view V (f) as a weighted polyhedral complex. Examples 1.2. Figures 1 through 5 give examples of two-variable tropical polynomials and their corresponding “zero loci.” All edges have weight 1 unless otherwise indicated. We also indicate the monomial determining the function on each domain of linearity and the precise position of the vertices. We now explain a simple way to see what V (f) looks like. Given f = n∈S anzn,

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