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20
SYZYGIES OF ORIENTED MATROIDS
 DUKE MATHEMATICAL JOURNAL VOL. 111, NO. 2
, 2002
"... We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids, and oriented matroids. These are StanleyReisner ideals of complexes of independent sets and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular rea ..."
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Cited by 15 (2 self)
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We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids, and oriented matroids. These are StanleyReisner ideals of complexes of independent sets and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of R. Stanley’s formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by D. Bayer, S. Popescu, and B. Sturmfels [3]. We resolve the combinatorial problems posed in [3] by computing Möbius invariants of graphic and cographic arrangements in terms of Hermite polynomials.
Tropical hyperplane arrangements and oriented matroids
 Math. Zeitschrift
"... Abstract. We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, ..."
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Cited by 12 (0 self)
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Abstract. We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection. 1.
Triangulations With Very Few Geometric Bistellar Neighbors
 Discrete Comput. Geom
, 1998
"... We are interested in a notion of elementary change between triangulations of a point configuration, the socalled bistellar flips, introduced by Gel'fand, Kapranov and Zelevinski. We construct sequences of triangulations of point configurations in dimension 3 with n 2 +2n+2 vertices and only 4n \G ..."
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Cited by 9 (3 self)
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We are interested in a notion of elementary change between triangulations of a point configuration, the socalled bistellar flips, introduced by Gel'fand, Kapranov and Zelevinski. We construct sequences of triangulations of point configurations in dimension 3 with n 2 +2n+2 vertices and only 4n \Gamma 3 geometric bistellar flips (for every even integer n), and of point configurations in dimension 4 with arbitrarily many vertices and a bounded number of flips. This drastically improves previous examples and seems to be evidence against the conjecture that any two triangulations of a point configuration can be joined by a sequence of flips. Introduction Given a finite point configuration A in the Euclidean space R d of dimension d we call triangulations of A all the geometrically realized simplicial complexes which cover the convex hull of A and which have their sets of vertices contained in A. In this paper we are interested in a notion of vicinity or elementary change between tri...
The Graph of Triangulations of a Point Configuration With D+4 Vertices is 3Connected.
, 1999
"... We study the graph of bistellar flips between triangulations of a vector configuration A with d + 4 elements in rank d + 1 (i.e. with corank 3), as a step in the Baues problem. We prove that the graph is connected in general and 3connected for acyclic vector configurations, which include all point ..."
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Cited by 9 (6 self)
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We study the graph of bistellar flips between triangulations of a vector configuration A with d + 4 elements in rank d + 1 (i.e. with corank 3), as a step in the Baues problem. We prove that the graph is connected in general and 3connected for acyclic vector configurations, which include all point configurations of dimension d with d + 4 elements. Hence, every pair of triangulations can be joined by a finite sequence of bistellar flips and every triangulation has at least 3 geometric bistellar neighbours. In corank 4, connectivity is not known and having at least 4 flips is false. In corank 2, the results are trivial since the graph is a cycle. Our methods are based in a dualization of the concept of triangulation of a point or vector configuration A to that of virtual chamber of its Gale transform B, introduced by de Loera et al. in 1996. As an additional result we prove a topological representation theorem for virtual chambers, stating that every virtual chamber of a rank 3 vector ...
The Generalized Baues Problem For Cyclic Polytopes
, 1998
"... The Generalized Baues Problem asks whether for a given point configuration the order complex of all its proper polyhedral subdivisions, partially ordered by refinement, is homotopy equivalent to a sphere. In this paper, an affirmative answer is given for the vertex sets of cyclic polytopes in all di ..."
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Cited by 9 (5 self)
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The Generalized Baues Problem asks whether for a given point configuration the order complex of all its proper polyhedral subdivisions, partially ordered by refinement, is homotopy equivalent to a sphere. In this paper, an affirmative answer is given for the vertex sets of cyclic polytopes in all dimensions. This yields the first nontrivial class of point configurations with neither a bound on the dimension, the codimension, nor the number of vertices for which this is known to be true. Moreover, it is shown that all triangulations of cyclic polytopes are lifting triangulations. This contrasts the fact that in general there are many nonregular triangulations of cyclic polytopes. Beyond this, we find triangulations of C 11 5 with flip deficiency. This proves—among other things—that there are triangulations of cyclic polytopes that are nonregular for every choice of points on the moment curve.
Flag arrangements and triangulations of products of simplices
 FORMAL POWER SERIES AND ALGEBRAIC COMBINATORICS
, 2006
"... ..."
On the Refinements of a Polyhedral Subdivision
 COLLECT. MATH
, 2000
"... Let : P ! Q be an affine projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by (or induced) and the fiber polytope of the projection: a polytope \Sigma(P; ) of dimension dim(P ) \Gamma dim(Q) whose faces a ..."
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Cited by 7 (4 self)
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Let : P ! Q be an affine projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by (or induced) and the fiber polytope of the projection: a polytope \Sigma(P; ) of dimension dim(P ) \Gamma dim(Q) whose faces are in correspondence with the coherent induced subdivisions (or coherent subdivisions). In this paper we investigate the structure of the poset of induced refinements of a induced subdivision. In particular, we define the refinement polytope associated to any induced subdivision S, which is a generalization of the fiber polytope and shares most of its properties. As applications of the theory we prove that if a point configuration has nonregular subdivisions then it has nonregular triangulations and we provide simple proofs of the existence of nonregular subdivisions for many particular point configurations.
Geometric bistellar flips: the setting, the context and a construction
 In International Congress of Mathematicians. Vol. III
, 2006
"... Abstract. We give a selfcontained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas. As a n ..."
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Cited by 5 (2 self)
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Abstract. We give a selfcontained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas. As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations. This shows, for the first time, that the poset of strict polyhedral subdivisions of a point set is not always connected.
Extremal Properties for Dissections of Convex 3Polytopes
 Siam J. Discrete Math
, 1999
"... A dissection of a convex dpolytope is a partition of the polytope into dsimplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the nu ..."
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Cited by 3 (2 self)
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A dissection of a convex dpolytope is a partition of the polytope into dsimplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of dsimplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3polytope and analyze extremal size triangulations for specific nonsimplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial dcubes.