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We define tropical analogues of the notions of linear space and Plücker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated dualization and transverse intersection to be constructible. Our main result that all constructible tropical linear spaces have(More)
In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus zero(More)
A Vinnikov curve is a projective plane curve which can be written in the form det(xX+yY +zZ) = 0 for X, Y and Z positive definite Hermitian n×n matrices. Given three n-tuples of positive real numbers, α, β and γ, there exist A, B and C ∈ GL n C with singular values α, β and γ and ABC = 1 if and only if there is a Vinnikov curve passing through the 3n points(More)
We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states of this recurrence are indexed by tilings of a polygon with(More)