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88
Lectures on 0/1Polytopes
, 1999
"... These lectures on the combinatorics and geometry of 0/1polytopes are meant as an introduction and invitation. Rather than heading for an extensive survey on 0/1polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1polytop ..."
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Cited by 16 (1 self)
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These lectures on the combinatorics and geometry of 0/1polytopes are meant as an introduction and invitation. Rather than heading for an extensive survey on 0/1polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1polytopes have a very simple definition and explicit descriptions; we can enumerate and analyze small examples explicitly in the computer (e. g. using polymake). However, any intuition that is derived from the analysis of examples in "low dimensions" will miss the true complexity of 0/1polytopes. Thus, in the following we will study several aspects of the complexity of higherdimensional 0/1polytopes: the doublyexponential number of combinatorial types, the number of facets which can be huge, and the coefficients of defining inequalities which sometimes turn out to be extremely large. Some of the effects and results will be backed by proofs in the course of these lectures; we will also be able to ve...
BASIC PROPERTIES OF CONVEX POLYTOPES
, 1997
"... Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) ..."
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Cited by 14 (2 self)
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Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial
Neighborly cubical polytopes
 Discrete & Computational Geometry
, 2000
"... Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose rskeleton is combinatorially equivalent to that of the convex dpolytope Cn d ndimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubical p ..."
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Cited by 14 (1 self)
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Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose rskeleton is combinatorially equivalent to that of the convex dpolytope Cn d ndimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubical polytope Cn d maximizes the fvector among all cubical (d − 1)spheres with 2n vertices. While we show that this is true for polytopal spheres if n ≤ d+1, we also give a counterexample for d = 4 and n = 6. Further, the existence of neighborly cubical polytopes shows that the graph of the ndimensional cube, where n ≥ 5, is “dimensionally ambiguous ” in the sense of Grünbaum. We also show that the graph of the 5cube is “strongly 4ambiguous”. In the special case d = 4, neighborly cubical polytopes have f3 = f0 4 log2 f0 4 vertices, so the facetvertex ratio f3/f0 is not bounded; this solves a problem of Kalai, Perles and Stanley studied by Jockusch.
Enumeration and random realization of triangulated surfaces.arXiv:math.CO
"... We give a complete enumeration of triangulated surfaces with 9 and 10 vertices. Moreover, we discuss how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly. 1 ..."
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Cited by 13 (8 self)
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We give a complete enumeration of triangulated surfaces with 9 and 10 vertices. Moreover, we discuss how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly. 1
Coefficients and roots of Ehrhart polynomials
 In Integer points in polyhedra—geometry, number theory, algebra, optimization, volume 374 of Contemp. Math
, 2005
"... Abstract. The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We also investigate the roots of Ehrhart polynomials ..."
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Cited by 13 (3 self)
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Abstract. The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of Ehrhart polynomials and relate them to known inequalities. We also investigate the roots of Ehrhart polynomials. We prove that for fixed d, there exists a bounded region of C containing all roots of Ehrhart polynomials of dpolytopes, and that all real roots of these polynomials lie in [−d, ⌊d/2⌋). In contrast, we prove that when the dimension d is not fixed the positive real roots can be arbitrarily large. We finish with an experimental investigation of the Ehrhart polynomials of cyclic polytopes and 0/1polytopes. 1.
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 12 (1 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
GALOIS GROUPS OF SCHUBERT PROBLEMS Via Homotopy Computation
, 2009
"... Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois group ..."
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Cited by 12 (9 self)
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Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3planes in C^8 meeting 15 fixed 5planes nontrivially is the full symmetric group S_6006.
Polyhedral conditions for the nonexistence of the MLE for hierarchical loglinear models
, 2006
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COMPUTING OPTIMAL MORSE MATCHINGS
"... Abstract. Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on c ..."
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Cited by 11 (0 self)
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Abstract. Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results. 1.