Tropical geometry studies piecewise-linear objects obtained from algebraic varieties by means of a valuation. Its basic arithmetic replaces ordinary addition and multiplication by
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With these operations, polynomial equations become piecewise-linear conditions. A tropical hypersurface is the set where the minimum in a tropical polynomial is attained by at least two terms. Tropicalization preserves enough information to address questions about algebraic curves, enumerative geometry, and the amoebas of complex varieties.
