The tropical semiring (R∪{∞}, min, +) replaces addition with min and multiplication with +. Tropical polynomials are piecewise-linear. Visualize tropical curves, Legendre transforms, and shortest-path algorithms.
Tropical semiring: (R∪{∞}, ⊕=min, ⊗=+). Addition: a⊕b=min(a,b). Multiplication: a⊗b=a+b. "0"=∞, "1"=0. Tropical polynomial: min of linear functions → piecewise linear (convex). Tropical curve: set where minimum achieved ≥2 times → piecewise-linear graph. Connection: amoebas of algebraic curves converge to tropical curves as Log→0. Shortest paths = tropical matrix multiplication (Dijkstra = tropical algebra). Connections: min-plus convolution, dynamic programming, idempotent analysis.