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This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank, and we show that they differ in general. Connections to polyhedral geometry, particularly to subdivisions of products of… (More)
Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results for the Markov bases, Graver bases and facet posets of these configurations, and we discuss applications to the statistical theory of log-linear models.
We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) l, we show that the numbers of regular and non-regular triangulations of ∆ l × ∆ k grow, respectively, as k Θ(k) and 2 Ω(k 2). For the special case of ∆ 2 × ∆ k , we relate triangulations to certain class of lozenge tilings. This allows us to… (More)
We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc. 13:3 (2000), 611–637 not only in size, but also in that their toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the… (More)
The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, that any two vertices of the polytope can be connected by a path of at most n − d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It… (More)
A pseudo-triangle is a simple polygon with exactly three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of… (More)
We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 n O(n −6) straight-edge triangulations of its convex hull. This improves the previous upper bound of 276.75 n+O(log(n)) .
By the \space of triangulations" of a nite point connguration A we mean either of the following two objects: the partially order set (poset) of all polyhedral subdivisions of A (the so-called Baues poset of A) considered as an abstract simplicial complex in the standard way or the graph of triangulations of A, whose vertices are the triangulations of A and… (More)