Related papers: Computations associated with the resonance arrange…
The resonance arrangement $\mathcal{A}_n$ is the arrangement of hyperplanes which has all non-zero $0/1$-vectors in $\mathbb{R}^n$ as normal vectors. It is the adjoint of the Braid arrangement and is also called the all-subsets arrangement.…
Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where…
An arrangement of hyperplanes is a finite collection of hyperplanes in a real Euclidean space. To such a collection one associates the characteristic polynomial that encodes the combinatorics of intersections of the hyperplanes. Finding the…
We classify one-element extensions of a hyperplane arrangement by the induced adjoint arrangement. Based on the classification, several kinds of combinatorial invariants including Whitney polynomials, characteristic polynomials, Whitney…
We give a complete formula for the characteristic polynomial of hyperplane arrangements $\mathcal J_n$ consisting of the hyperplanes $x_i+x_j=1$, $x_k=0$, $x_l=1$, $ 1\leq i, j, k, l\leq n$. The formula is obtained by associating hyperplane…
A hyperplane arrangement in $\mathbb{R}^n$ is a finite collection of affine hyperplanes. The regions are the connected components of the complement of these hyperplanes. By a theorem of Zaslavsky, the number of regions of a hyperplane…
We study the connection between triangulations of a type $A$ root polytope and the resonance arrangement, a hyperplane arrangement that shows up in a surprising number of contexts. Despite an elementary definition for the resonance…
Characteristic elements of the Tits algebra of a real hyperplane arrangement carry information about the characteristic polynomial. We present this notion and its basic properties, and apply it to derive various results about the…
If \A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=\pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\k), viewed as a…
Let A be a subspace arrangement and let chi(A,t) be the characteristic polynomial of its intersection lattice L(A). We show that if the subspaces in A are taken from L(B_n), where B_n is the type B Weyl arrangement, then chi(A,t) counts a…
We give a formula for computing the characteristic polynomial for certain hyperplane arrangements in terms of the number of bipartite graphs of given rank and cardinality.
We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this…
Let $\mathcal{A}$ be a central hyperplane arrangement in $\mathbb{C}^{n+1}$ and $H_i,i=1,2,...,d$ be the defining equations of the hyperplanes of $\mathcal{A}$. Let $f=\prod_i H_i$. There is a global Milnor fibration $F\hookrightarrow…
We introduce a new algorithm computing the characteristic polynomials of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its…
Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic…
If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\k)$ is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $\phi_k$, of the lower central series quotients of $\pi_1(M)$ can be computed from the…
We study the expected behavior of the Betti numbers of arrangements of the zeros of random (distributed according to the Kostlan distribution) polynomials in $\mathbb{R}\mathrm{P}^n$. Using a random spectral sequence, we prove an…
Kamiya, Takemura, and Terao introduced a characteristic quasi-polynomial which enumerates the numbers of elements in the complement of hyperplane arrangements modulo positive integers. In this paper, we compute the characteristic…
Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic…
An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for the hyperplanes. We study the…