Tropical Variety

The tropical hypersurface of a tropical polynomial (see tropical-semiring) is its corner locus, the set of points where the minimum is attained by at least two terms at once,

Away from a single affine term dominates and is smooth, whereas on the graph creases. It is a polyhedral complex of pure dimension , with flat facets meeting along lower-dimensional ridges. A tropical variety is a finite intersection of such hypersurfaces.

Two structural facts make the complex rigid, not arbitrary:

• Newton duality. is dual to the regular subdivision of the newton-polytope induced by the coefficients . Each facet of is perpendicular to an edge of the subdivision, so the coefficients, acting as lifting heights, determine exactly which facets appear and how they meet.
• Balancing. Weight each facet by the lattice length of its dual edge. Around every ridge the primitive facet normals, so weighted, sum to zero. This balancing condition is the tropical shadow of a conservation law, and it is why the facets hang together like a crystal rather than floating free.

Tropical varieties arise as degenerations. The tropicalisation of an algebraic variety over a valued field is the limit of its amoeba, the image under coordinatewise , so is the skeleton of an honest variety. Rigorous meaning survives the projection to a piecewise-linear form. Rendered as a faceted solid, the same corner locus carves the crystal forms of quartz-and-obsidian-tropically.