Tropical Geometry: Piecewise-Linear Curves

Tropical algebra: a ⊕ b = max(a,b), a ⊗ b = a+b. Polynomials become piecewise-linear functions.

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Tropical algebra: Replace (ℝ,+,×) with (ℝ∪{−∞}, max, +). Monomials aᵢⱼ + i·x + j·y compete — the tropical polynomial is their pointwise maximum.
Tropical curve: The locus where ≥2 monomials simultaneously achieve the maximum. Always a piecewise-linear graph (a tropical variety).
Tropical Bézout: Two tropical curves of degrees d₁, d₂ intersect in exactly d₁·d₂ points (counted with multiplicity) — same as classical!
Newton polygon: The regular subdivision of the Newton polygon of f gives the combinatorial structure of the tropical curve.