Related papers: Localization of plus-one generated arrangements
In $d$-dimensional space (over any field), given a set of lines, a joint is a point passed through by $d$ lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by $L$ lines, and it…
We construct counterexamples to Yoshinaga's conjecture that every free arrangement is either inductively free or rigid in characteristic zero. The smallest example has $13$ hyperplanes, its intersection lattice has a one dimensional moduli…
We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms.…
The class of subarrangements of the well-studied braid arrangement, so-called graphic hyperplane arrangements, is important for analysing new concepts and properties in hyperplane arrangement theory. While there is a nice characterization…
We study the number of hamiltonian circuits, containing a fixed basis, and the number of hyperplanes, which do not contain a fixed basis in perfect matroid designs. Projective and affine finite geometries are considered as examples of such…
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras,…
We define and study the Tutte polynomial of a hyperplane arrangement. We introduce a method for computing it by solving an enumerative problem in a finite field. For specific arrangements, the computation of Tutte polynomials is then…
A modified Lagrange Polynomial is introduced for polynomial extrapolation, which can be used to estimate the equally spaced values of a polynomial function. As an example of its application, this article presents a prime-generating…
We study the notion of a nice partition or factorization of a hyperplane arrangement due to Terao from the early 1990s. The principal aim of this note is an analogue of Terao's celebrated addition-deletion theorem for free arrangements for…
We study bipartite maps on the plane with one infinite face and one face of perimeter 2. At first we consider the problem of their enumeration an then study the connection between the combinatorial structure of a map and the degree of its…
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study…
We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when…
We show that if the fundamental groups of the complements of two line arrangements in the complex projective plane are isomorphic to the same direct sum of free groups, then the complements of the arrangements are homotopy equivalent. For…
In the present note we study determinantal arrangements constructed with use of the $3$-minors of a $3 \times 5$ generic matrix of indeterminates. In particular, we show that certain naturally constructed hypersurface arrangements in…
In this paper, we propose a new method for localization of polynomial ideal, which we call "Local Primary Algorithm". For an ideal $I$ and a prime ideal $P$, our method computes a $P$-primary component of $I$ after checking if $P$ is…
We give a combinatorial characterization of isotropic subspaces in the Orlik- Solomon algebra of a hyperplane arrangement in terms of decorations of its intersection lattice. We then use this characterization to prove a result that relates…
We study enumerative questions on the moduli space $\mathcal{M}(L)$ of hyperplane arrangements with a given intersection lattice $L$. Mn\"ev's universality theorem suggests that these moduli spaces can be arbitrarily complicated; indeed it…
We give explicit generators for ideals of two classes of subspace arrangements embedded in certain reflection arrangements, generalizing results of Li-Li and Kleitman-Lovasz. We also give minimal generators for the ideals of arrangements…
In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated…
A central arrangement $\A$ of hyperplanes in an $\ell$-dimensional vector space $V$ is said to be {\it totally free} if a multiarrangement $(\A, m)$ is free for any multiplicity $ m : \A\to \Z_{> 0}$. It has been known that $\A$ is totally…