Min-plus algebra: tropical lines, conics, Newton polygons, piecewise-linear varieties
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Coefficients (min-plus: f = min(a_{ij} + ix + jy))
Tropical curve V(f)
Newton polygon & dual subdivision
Tropical semiring: (ℝ∪{∞}, min, +). Addition → min, multiplication → +.
A tropical polynomial f(x,y) = min_{i,j}(a_{ij} + ix + jy) is a piecewise-linear convex function.
The tropical variety V(f) = {(x,y) : minimum is attained at least twice} — a balanced weighted graph (Mikhalkin 2003).
Dual to V(f): a subdivision of the Newton polygon. Tropical curves count real/complex curves via Gromov-Witten theory.