1.2. SOME BACKGROUND ON FANS 11 Figure 13. Two tropical conics meeting at four points. Here m1,m2 ∈ M ∼ Z2, and 2 M ∼ Z, so |m1 ∧ m2| makes sense as a positive number no matter which isomorphism is chosen. We then have the tropical B´ezout theorem, which states that P ∈C∩D iP (C, D) = d · e. This is exactly the expected result for ordinary algebraic curves in P2, of course. For a proof, see [96], §4. See Figure 13 for an example. 1.2. Some background on fans We will collect here a number of standard notions concerning fans. We send the reader to [27] for more details. Definition 1.7. A strictly convex rational polyhedral cone in MR is a lattice polyhedron in MR with exactly one vertex, which is 0 ∈ MR. A fan Σ in MR is a set of strictly convex rational polyhedral cones such that (1) If σ ∈ Σ, and τ ⊆ σ is a face, then τ ∈ Σ. (2) If σ1,σ2 ∈ Σ, then σ1 ∩ σ2 is a face of σ1 and σ2. In other words, a fan Σ is a polyhedral decomposition of a set |Σ| ⊆ MR, called the support of Σ, with all elements of the polyhedral decomposition being strictly convex rational polyhedral cones. A fan is complete if |Σ| = MR. Definition 1.8. Let Σ be a fan in MR. A PL (piecewise linear) function on Σ is a continuous function ϕ : |Σ| → R which is linear when restricted to each cone of Σ. The function ϕ is strictly convex if (1) |Σ| is a convex set in MR (2) For m, m ∈ |Σ|, ϕ(m) + ϕ(m ) ≥ ϕ(m + m ), with equality holding if and only if m, m lie in the same cone of Σ.

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