Resurgences for ideals of special point configurations in PN coming from hyperplane arrangements

Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof). For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that states a linear bound for the diameter is possible. On the positive side, we show the conjecture is true for oriented matroids of low rank and low corank, and, verified with computers, for all oriented matroids with up to nine elements. On the negative side, our computer search showed two natural strengthenings of the main conjecture are false.

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Combinatorial Generation of Small Point Configurations and Hyperplane Arrangements

Algorithms and Combinatorics - Discrete and Computational Geometry ◽

10.1007/978-3-642-55566-4_19 ◽

2003 ◽

pp. 425-440 ◽

Cited By ~ 6

Author(s):

Lukas Finschi ◽

Komei Fukuda

Keyword(s):

Hyperplane Arrangements ◽

Point Configurations ◽

Combinatorial Generation ◽

Small Point

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Piles of Cubes, Monotone Path Polytopes, and Hyperplane Arrangements

Discrete & Computational Geometry ◽

10.1007/pl00009404 ◽

1999 ◽

Vol 21 (1) ◽

pp. 117-130 ◽

Cited By ~ 1

Author(s):

C. A. Athanasiadis

Keyword(s):

Hyperplane Arrangements ◽

Monotone Path

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Conditional probabilities via line arrangements and point configurations

Linear and Multilinear Algebra ◽

10.1080/03081087.2021.1912693 ◽

2021 ◽

pp. 1-33

Author(s):

Oliver Clarke ◽

Fatemeh Mohammadi ◽

Harshit J. Motwani

Keyword(s):

Conditional Probabilities ◽

Line Arrangements ◽

Point Configurations

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On polytopes and generalizations of the KLT relations

Journal of High Energy Physics ◽

10.1007/jhep12(2020)057 ◽

2020 ◽

Vol 2020 (12) ◽

Cited By ~ 1

Author(s):

Nikhil Kalyanapuram

Keyword(s):

Scattering Amplitudes ◽

Large Class ◽

Moduli Space ◽

Differential Forms ◽

Natural Generalization ◽

Intersection Theory ◽

Hyperplane Arrangements ◽

Number Of Solutions ◽

Scalar Theories ◽

Double Copy

Abstract We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.

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The Tutte polynomial of symmetric hyperplane arrangements

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500042 ◽

2020 ◽

Vol 29 (03) ◽

pp. 2050004

Author(s):

Hery Randriamaro

Keyword(s):

Hyperplane Arrangement ◽

Dual Graph ◽

Tutte Polynomial ◽

Hyperplane Arrangements ◽

Weyl Groups ◽

Reflection Groups ◽

Tutte Polynomials ◽

One Year ◽

The One ◽

Definition Of

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

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Elementary Notes on Simple Summations and Summations of Products

Transactions of the Faculty of Actuaries ◽

10.1017/s0071368600002342 ◽

1936 ◽

Vol 15 (1) ◽

pp. 141-176

Author(s):

Duncan C. Fraser

Keyword(s):

Orthogonal Polynomials ◽

Least Squares ◽

Divided Differences ◽

Binomial Coefficients ◽

Special Point

SynopsisThe paper is intended as an elementary introduction and companion to the paper on “Orthogonal Polynomials,” by G. J. Lidstone, J.I.A., vol. briv., p. 128, and the paper on the “Sum and Integral of the Product of Two Functions,” by A. W. Joseph, J.I.A., vol. lxiv., p. 329; and also to Dr. Aitken's paper on the “Graduation of Data by the Orthogonal Polynomials of Least Squares,” Proc. Roy. Soc. Edin., vol. liii., p. 54.Following Dr. Aitken Σux is defined for the immediate purpose to be u0+…+ux−1.The scheme of successive summations is set out in the form of a difference diagram and is extended to negative arguments. The special point to which attention is drawn is the existence of a wedge of zeros between the sums for positive arguments and those for negative arguments.The rest of the paper is for the greater part a study of the table of binomial coefficients for positive and for negative arguments. The Tchebychef polynomials are simple functions of the binomial coefficients, and after a description of a particular example and of its properties general methods are given of forming the polynomials by means of tables of differences. These tables furnish examples of simple, differences, of divided differences, of adjusted differences, and of a system of special adjusted differences which gives a very easy scheme for the formation of the Tchebychef polynomials.

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