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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)

- Francisco Santos, Bernd Sturmfels
- J. Comb. Theory, Ser. A
- 2003

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.

- Francisco Santos
- ArXiv
- 2010

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)

- Ruth Haas, David Orden, +6 authors Walter Whiteley
- Symposium on Computational Geometry
- 2003

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with <i>pointed</i> vertices (incident to an angle larger than <i>p</i>). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial… (More)

We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of the point set and whose edges are flips of… (More)

- Francisco Santos, Raimund Seidel
- J. Comb. Theory, Ser. A
- 2003

We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 O(n−6) straight-edge triangulations of its convex hull. This improves the previous upper bound of 276.75n+O(log(n)).

- Francisco Santos
- 1997

We consider the concept of triangulation of an oriented matroid. We provide a de nition which generalizes the previous ones by Billera{ Munson and by Anderson and which specializes to the usual notion of triangulation (or simplicial fan) in the realizable case. Then we study the relation existing between triangulations of an oriented matroidM and extensions… (More)

- Francisco Santos
- 2005

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 ∆ ×∆ grow, respectively, as k and 2 2). For the special case of ∆ ×∆, we relate triangulations to certain class of lozenge tilings. This allows us to compute the exact number… (More)

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n− d. That is to say, we can go from any vertex to any other vertex using at most n− d edges. Despite being one of the most fundamental, basic and old problems in… (More)