Tropical Combinatorics

A Tropical Hypersurface
point vi ∈ T d−1 = apex of unique max-tropical hyperplane
H max (vi)
homogeneous linear form hi ∈ C{{z}}[x1, xx, . . .,xd];
h := h1 · h2 · · ·hn
Proposition
The tropical hypersurface defined by trop max (h) is the union of
the max-tropical hyperplanes in A.
Corollary
Let p ∈ T d−1 \ A be a generic point. Then its coarse type tA(p)
equals the exponent of the monomial in h which defines the unique
facet of P(trop max (h)) above p.