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165
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 28 (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
Lectures on 0/1polytopes
 Polytopes — combinatorics and computation (Oberwolfach, 1997), volume 29 of DMV Seminar
, 2000
"... 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. 0/1polytope ..."
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Cited by 25 (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 verify some of them on explicit examples, which are accessible as a polymake database.
The hyperdeterminant and triangulations of the 4cube
, 2008
"... The hyperdeterminant of format 2 × 2 × 2 × 2isapolynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4cube. The 87959448 regular triangulations of the 4cube are classified into 25448 Dequivalence class ..."
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Cited by 25 (4 self)
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The hyperdeterminant of format 2 × 2 × 2 × 2isapolynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4cube. The 87959448 regular triangulations of the 4cube are classified into 25448 Dequivalence classes, one for each vertex of the Newton polytope. The 4cube has 80876 coarsest regular subdivisions, one for each facet of the secondary polytope, but only 268 of them come from the hyperdeterminant.
The Newton polytope of the implicit equation
 Moscow Math. J
"... Abstract. We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope. ..."
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Cited by 25 (2 self)
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Abstract. We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.
Coefficients and roots of Ehrhart polynomials
 IN INTEGER POINTS IN POLYHEDRA—GEOMETRY, NUMBER THEORY, ALGEBRA, OPTIMIZATION, VOLUME 374 OF CONTEMP. MATH
, 2005
"... 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 pro ..."
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Cited by 25 (3 self)
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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.
Computing Maximum Likelihood Estimates in loglinear models
, 2006
"... We develop computational strategies for extended maximum likelihood estimation, as defined in Rinaldo (2006), for general classes of loglinear models of widespred use, under Poisson and productmultinomial sampling schemes. We derive numerically efficient procedures for generating and manipulating ..."
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Cited by 25 (4 self)
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We develop computational strategies for extended maximum likelihood estimation, as defined in Rinaldo (2006), for general classes of loglinear models of widespred use, under Poisson and productmultinomial sampling schemes. We derive numerically efficient procedures for generating and manipulating design matrices and we propose various algorithms for computing the extended maximum likelihood estimates of the expectations of the cell counts. These algorithms allow to identify the set of estimable cell means for any given observable table and can be used for modifying traditional goodnessoffit tests to accommodate for a nonexistent MLE. We describe and take advantage of the connections between extended maximum likelihood
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 23 (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.
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 22 (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 grou ..."
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Cited by 21 (18 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 S6006.