8 1. TROPICS 0 0 Figure 8. The Newton polytope and subdivision for Figure 1. In fact, ˇ can be described in a more familiar way. Note that ˇ(m) = min{ϕ(n) + m, n| n ∈ ΔS}. From this, it is clear that ˇ max consists of the maximal domains of linearity of ˇ, with ˇ|ˇ having slope v for v a vertex (element of P[0]) of P. Indeed, the minimum is always achieved at some vertex, and if this vertex is v, then ˇ(m) = ϕ(v) + m, v . Thus ˇ is linear on ˇ with slope v. Furthermore, as necessarily ϕ(v) + m, v≤ ϕ(v ) + m, v whenever m ∈ ˇ, one sees that ˇ is in fact given by the tropical polynomial n∈P[0] ϕ(n)zn. This is not necessarily the original polynomial defining the function f. However, clearly the vertices of ˜ S are of the form (n, an) for n ∈ P[0] ⊆ S, so ϕ(n) = an for n ∈ P[0], and the tropical polynomial defining ˇ is simply missing some of the terms of the original defining polynomial f. These missing terms are precisely ones of the form anzn with (n, an) not a vertex of ˜ S . We can see that such terms are irrelevant for calculating f. Indeed, if (n, an) is not a vertex of ˜ S for some n ∈ N ∩ ΔS, and f(m) = m, n + an for some m ∈ MR, then m, n + an ≤ m, n + an for all n ∈ S. But then the hyperplane in NR × R given by {(n , r) ∈ NR × R | m, n + r = m, n + an} is a supporting hyperplane for ˜ S which contains (n, an), and hence must also contain a vertex (n , an ) of ˜ S . Then f(m) coincides with m, n +an , and hence the term anzn was irrelevant for calculating f. Thus we see ˇ = f. Since the domains of linearity of ˇ are the polyhedra of ˇ max , we see that V (f) = τ∈P[1] ˇ Since the 0-cells of ˇ are the cells ˇ = {−mσ} for σ ∈ Pmax, it is usally easy to draw V (f) using this description. Additionally, the weights are easily determined: for τ ∈ P[1], the weight of ˇ is just the aﬃne length of τ, i.e., the index of the difference of the endpoints of τ. To summarize, the function ϕ determines the dual decomposition ˇ, whose vertices are given by slopes of ϕ, and V (f) is the codimension one skeleton of ˇ. Examples 1.5. For the examples of Figures 1 through 5, the Newton polytopes along with their regular decomposition and values of an are given in Figures 8 throught 12.

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