12 1. THE TROPICS The function ϕ is integral if for each σ ∈ Σmax there exists an nσ ∈ N such that nσ and ϕ agree on σ. The Newton polyhedron of a strictly convex PL function ϕ : |Σ| → R is Δϕ := {n ∈ NR | ϕ(m) + n, m≥ 0 for all m ∈ |Σ|}. The Newton polyhedron of a function ϕ is unbounded if and only if Σ is not a complete fan. If Σ is complete, it is easy to see that Δϕ = Conv({−nσ | σ ∈ Σmax}) where nσ ∈ NR is the linear function ϕ|σ. Note there is a one-to-one inclusion reversing correspondence between cones in Σ and faces of Δϕ, with σ ∈ Σ corre- sponding to {n ∈ Δϕ | ϕ(m) + n, m = 0 for all m ∈ σ}. Definition 1.9. If Δ ⊆ NR is a polyhedron, σ ⊆ Δ a face, the normal cone to Δ along σ is NΔ(σ) = {m ∈ M | m|σ = constant, m, n≥ m, n for all n ∈ Δ, n ∈ σ}. If τ ⊆ σ is a subset, then Tτσ denotes the tangent wedge to σ along τ, defined by Tτσ = {r(m − m ) | m ∈ σ, m ∈ τ, r ≥ 0}. The normal fan of Δ is ˇ Δ := {NΔ(σ) | σ is a face of Δ}. One checks easily that (1.3) TσΔ = (NΔ(σ))∨ := {n ∈ N | n, m≥ 0 ∀m ∈ NΔ(σ)}. The normal fan ˇ Δ to Δ carries a PL function ϕΔ : |ˇ Δ | → R defined by ϕΔ(m) = − inf{n, m| n ∈ Δ}. It is easy to see that if ϕ : |Σ| → R is strictly convex, then Σ is the normal fan to Δϕ. This in fact gives a one-to-one correspondence between strictly convex PL functions ϕ on a fan Σ and polyhedra Δ with normal fan Σ. Note given Δ, Δϕ Δ = Δ, and given ϕ : |Σ| → R, ϕΔ ϕ = ϕ. Definition 1.10. If Σ is a fan in MR, τ ∈ Σ, we define the quotient fan Σ(τ) of Σ along τ to be the fan Σ(τ) := {(σ + Rτ)/Rτ | σ ∈ Σ,τ ⊆ σ} in MR/Rτ, where Rτ is the linear space spanned by τ. If ϕ : |Σ| → R is a PL function on Σ, then ϕ induces a function ϕ(τ) : Σ(τ) → R, well-defined up to linear functions, as follows. Choose n ∈ NR such that ϕ(m) = n, m for m ∈ τ. Then for any σ containing τ, ϕ|σ −n is zero on τ, hence descends to a linear function on (σ + Rτ)/Rτ. These piece together to give a PL function ϕ(τ) on Σ(τ), well-defined up to a linear function (determined by the choice of n). Note that if ϕ is strictly convex, then so is ϕ(τ). If M R = MR/Rτ and N R = Hom(MR, R), then NR = (Rτ)⊥. It is then easy to see that Δϕ(τ) is just the translate by n of the face of Δϕ corresponding to τ.

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