Locally heavy hyperplanes in multiarrangements

Hyperplane arrangements of rank 3 admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note we prove that arrangements which admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary we give a new proof that irreducible arrangements with a generic hyperplane are totally nonfree. In another application we show that an irreducible multiarrangement of rank 3 with at least two locally heavy hyperplanes is not free.     

The Euler multiplicity and addition-deletion theorems for multiarrangements

The addition–deletion theorems for hyperplane arrangements,which were originally shown by Terao [J. Fac. Sci. Univ. TokyoSect. IA Math. 27 (1980) 293–320.], provide useful waysto construct examples of free arrangements. In this article,we prove addition–deletion theorems for multiarrangements.A key to the generalization is the definition of a new multiplicity,called the Euler multiplicity, of a restricted multiarrangement.We compute the Euler multiplicities in many cases. Then we applythe addition–deletion theorems to various arrangements,including supersolvable arrangements and the Coxeter arrangementof type A₃, to construct free and non-free multiarrangements.     

Derivation radical subspace arrangements

In this note we study modules of derivations on collections of linear subspaces in a finite dimensional vector space. The central aim is to generalize the notion of freeness from hyperplane arrangements to subspace arrangements. We call this generalization ‘derivation radical’. We classify all coordinate subspace arrangements that are derivation radical and show that certain subspace arrangements of the Braid arrangement are derivation radical. We conclude by proving that under an algebraic condition the subspace arrangement consisting of all codimension c intersections, where c is fixed, of a free hyperplane arrangement are derivation radical.     

Characterization of a free arrangement and conjecture of Edelman and Reiner

We consider a hyperplane arrangement in a vector space of dimension four or higher. In this case, the freeness of the arrangement is characterized by properties around a fixed hyperplane. As an application, we prove the freeness of cones over certain truncated affine Weyl arrangements which was conjectured by Edelman and Reiner.     

Free arrangements and coefficients of characteristic polynomials

Ziegler showed that the multirestriction of a free arrangement is also free. After Ziegler’s work, several results concerning the “reverse direction”, i.e., characterizing freeness of an arrangement via that of its multirestriction, have appeared. In this paper, we prove a new characterization of freeness in which the second Betti number of the arrangement plays a crucial role.     

On a Generalization of Zaslavsky’s Theorem for Hyperplane Arrangements

We define arrangements of codimension-1 submanifolds in a smooth manifold which generalize arrangements of hyperplanes. When these submanifolds are removed the manifold breaks up into regions, each of which is homeomorphic to an open disc. The aim of this paper is to derive formulas that count the number of regions formed by such an arrangement. We achieve this aim by generalizing Zaslavsky’s theorem to this setting. We show that this number is determined by the combinatorics of the intersections of these submanifolds.     

Asymptotically liberating sequences of random unitary matrices

A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting “asymptotic liberation” under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the second-named author and of Tulino, Caire, Shamai and Verdú.     

On the freeness of Böröczky line arrangements

Periodica Mathematica Hungarica - In the present note, we focus on the freeness and some combinatorial properties of line arrangements in the projective plane having only double and triple points....     

The Stability of the Family of B 2-Type Arrangements

We introduce the family of B ₂-type arrangements as a generalization of the classical Coxeter arrangement of type B ₂ and consider the stability and the freeness of it. We show the freeness and (semi)stability are determined by the combinatorics. Moreover, we give a partial answer to the 4-shift problem, which is a conjecture on the combinatorics and geometry induced from the B ₂-type arrangements.     

Freeness for 13 lines arrangements is combinatorial

We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane.     

Matricial R-transform

We study the addition problem for strongly matricially free random variables which generalize free random variables. Using operators of Toeplitz type, we derive a linearization formula for the matricial R-transform related to the associated convolution. It is a linear combination of Voiculescu?s R-transforms in free probability with coefficients given by internal units of the considered array of subalgebras. This allows us to view this formula as the matricial linearization property of the R-transform. Since strong matricial freeness unifies the main types of noncommutative independence, the matricial R-transform plays the role of a unified noncommutative analog of the logarithm of the Fourier transform for free, boolean, monotone, orthogonal, s-free and c-free independence.     

Affine and Toric Hyperplane Arrangements

We extend the Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky’s fundamental results on the number of regions.     

Generalized splines and graphic arrangements

We define a chain complex for generalized splines on graphs, analogous to that introduced by Billera and refined by Schenck–Stillman for splines on polyhedral complexes. The hyperhomology of this chain complex yields bounds on the projective dimension of the ring of generalized splines. We apply this construction to the module of derivations of a graphic multi-arrangement, yielding homological criteria for bounding its projective dimension and determining freeness. As an application, we show that a graphic arrangement admits a free constant multiplicity if and only if it splits as a product of braid arrangements.     

Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the n-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement.     

On the Freeness of 3-Arrangements

Hyperplane arrangements in a three-dimensional vector spaceare considered in this paper. A characterization of the freenessof such an arrangement is given in terms of the characteristicpolynomial and a restricted multiarrangement. As an application,the freeness of cones over certain two-dimensional affine arrangementsis proved. 2000 Mathematics Subject Classification 52C35 (primary),32S22 (secondary).     

Generalized Characteristics for Meromorphic in the Half-plane Functions

In this paper we introduce generalized characteristics for meromorphic in the halfplane functions, and generalize the Levin’s formula and the first fundamental theorem for Tsuji’s characteristics.     

Modules of Differential Operators of Order 2 on Coxeter Arrangements

The collection of reflection hyperplanes of a finite reflection group is called a Coxeter arrangement. A Coxeter arrangement is known to be free. K. Saito has constructed a basis consisting of invariant elements for the module of derivations on a Coxeter arrangement. We study the module of \(\mathcal{A}\) -differential operators as a generalization of the study of the module of \(\mathcal{A}\) -derivations. In this article, we prove that the modules of differential operators of order 2 on Coxeter arrangements of types A, B and D are free, by exhibiting their bases. We also prove that the modules cannot have bases consisting of only invariant elements. Two keys for the proof of freeness are the “Cauchy-Sylvester theorem on compound determinants” and the “Saito-Holm criterion for freeness.”     

Rees–Shishikura’s theorem for geometrically finite rational maps

In this paper, we generalize Rees–Shishikura’s theorem to the class of geometrically finite rational maps.     

Refined theorems of approximation theory in the space of p-absolutely continuous functions

In this paper, we prove direct and inverse theorems of approximation theory in the space of p-absolutely continuous functions which generalize Terekhin’s results in the same way as Timan’s results in L _p generalize the classical theorems of approximation theory. The main theorems are refined for functions with quasimonotone Fourier coefficients and, in a number of cases, the resulats are shown to be sharp.     

On the Sylvester–Gallai and the orchard problem for pseudoline arrangements

We study a non-trivial extreme case of the orchard problem for 12 pseudolines and we provide a complete classification of pseudoline arrangements having 19 triple points and 9 double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of Böröczky. Since Melchior’s inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Grünbaum’s problems. We formulate some open problems which involve our new examples of line arrangements.