f(x,y) = max(2x, x+y+a, 2y+b, x+c, y, 0)
Tropical semiring: (ℝ∪{−∞}, ⊕, ⊗) where a⊕b = max(a,b), a⊗b = a+b.
Tropical curves = piecewise-linear skeleta (1-skeleton of Newton subdivision). A degree-d curve has d(d+3)/2 monomials.
Dequantization: aᵦ⊕bᵦ = ℏ·log(e^{a/ℏ}+e^{b/ℏ}) → max(a,b) as ℏ→0.
Applications: shortest paths, phylogenetics, mirror symmetry, VLSI circuit design.