1.3. PARAMETERIZED TROPICAL CURVES 15 ρ1 ρ2 ρ0 Figure 14. The fan for P2. we need to assign a weight w(E) to each edge E of h(Γ). We define this as follows. Pick a point m ∈ E which is not a vertex of h(Γ) and is not the image of any vertex of Γ, and define w(E) = E ∈Γ[1] E ∩h−1(m)=∅ w(E ), i.e., the weight of E is the sum of weights of edges of Γ whose image under h contains m. It is easy to check that the balancing condition on h implies firstly that this weight is well-defined, i.e., doesn’t depend on the choice of m, and secondly that h(Γ) satisfies the balancing condition. Proposition 1.15. If h : Γ → MR is a tropical curve with dim MR = 2, then there exists a tropical polynomial f such that h(Γ) = V (f), as weighted one- dimensional polyhedral complexes. Proof. We define f as follows. h(Γ) yields a polyhedral decomposition ˇ of MR whose maximal cells are closures of connected components of MR \ h(Γ). Choose some cell σ0 ∈ ˇ max and define f|σ 0 ≡ 0. We then define f inductively. Suppose f is defined on σ ∈ ˇ max . If σ ∈ ˇ max and E = σ∩σ satisfies dim E = 1, then we can define f on σ as follows. Extend f|σ to an aﬃne linear function fσ : MR → R. Let nE ∈ N be a primitive normal vector to E which takes a constant value on E and takes larger values on σ than on σ . Denote by nE,E the value nE takes on E. Then we define f to be fσ + w(E)(nE − nE,E ) on σ . It is an immediate consequence of the balancing condition that this is well-defined. Indeed, if we define f on a sequence of polygons with a common vertex V , starting at σ and passing successively through edges E1,...,En, then when we return to σ, we have constructed the function fσ + ∑ n i=1 w(Ei)(nE i − nE i , Ei ) = fσ by the balancing condition.

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