Editing Arrangement of hyperplanes (section)

== Real arrangements ==

In [[real number|real]] [[affine space]], the complement is disconnected: it is made up of separate pieces called '''cells''' or '''regions''' or '''chambers''', each of which is either a bounded region that is a [[Convex polygon|convex]] [[polytope]], or an unbounded region that is a convex [[polyhedron#General|polyhedral]] region which goes off to infinity.  
Each flat of ''A'' is also divided into pieces by the hyperplanes that do not contain the flat; these pieces are called the '''faces''' of ''A''.  
The regions are faces because the whole space is a flat.  
The faces of codimension 1 may be called the '''facets''' of ''A''.  
The '''face semilattice''' of an arrangement is the set of all faces, ordered by ''inclusion''.  Adding an extra top element to the face semilattice gives the '''face lattice'''.

In two dimensions (i.e., in the real affine [[plane (mathematics)|plane]]) each region is a convex [[polygon]] (if it is bounded) or a convex polygonal region which goes off to infinity.  
* As an example, if the arrangement consists of three parallel lines, the intersection semilattice consists of the plane and the three lines, but not the empty set.  There are four regions, none of them bounded.  
* If we add a line crossing the three parallels, then the intersection semilattice consists of the plane, the four lines, and the three points of intersection.  There are eight regions, still none of them bounded.  
* If we add one more line, parallel to the last, then there are 12 regions, of which two are bounded [[parallelogram]]s.

Typical problems about an arrangement in ''n''-dimensional real space are to say how many regions there are, or how many faces of dimension 4, or how many bounded regions.  These questions can be answered just from the intersection semilattice.  For instance, two basic theorems, from Zaslavsky (1975), are that the number of regions of an affine arrangement equals (&minus;1)<sup>''n''</sup>''p''<sub>''A''</sub>(&minus;1) and the number of bounded regions equals (&minus;1)<sup>''n''</sup>p<sub>''A''</sub>(1).  Similarly, the number of ''k''-dimensional faces or bounded faces can be read off as the coefficient of ''x''<sup>''n''&minus;''k''</sup> in (&minus;1)<sup>''n''</sup> w<sub>''A''</sub> (&minus;''x'', &minus;1) or (&minus;1)<sup>''n''</sup>''w''<sub>''A''</sub>(&minus;''x'', 1).

{{harvtxt|Meiser|1993}} designed a fast algorithm to determine the face of an arrangement of hyperplanes containing an input point.

Another question about an arrangement in real space is to decide how many regions are [[simplex|simplices]] (the ''n''-dimensional generalization of [[triangle]]s and [[tetrahedron|tetrahedra]]).  This cannot be answered based solely on the intersection semilattice. The [[McMullen problem]] asks for the smallest arrangement of a given dimension in general position in [[real projective space]] for which there does not exist a cell touched by all hyperplanes.

A real linear arrangement has, besides its face semilattice, a '''[[poset]] of regions''', a different one for each region.  This poset is formed by choosing an arbitrary base region, ''B''<sub>0</sub>, and associating with each region ''R'' the set ''S''(''R'') consisting of the hyperplanes that separate ''R'' from ''B''. The regions are partially ordered so that ''R''<sub>1</sub> ≥ ''R''<sub>2</sub> if ''S''(''R''<sub>1</sub>, ''R'') contains ''S''(''R''<sub>2</sub>, ''R'').  In the special case when the hyperplanes arise from a [[root system]], the resulting poset is the corresponding [[Weyl group]] with the weak order. In general, the poset of regions is [[ranked poset|ranked]] by the number of separating hyperplanes and its [[Incidence algebra|Möbius function]] has been computed {{harv|Edelman|1984}}.

Vadim Schechtman and [[Alexander Varchenko]] introduced a matrix indexed by the regions. The matrix element for the region <math>R_i</math> and <math>R_j</math> is given by the product of indeterminate variables <math>a_H</math> for every hyperplane H that separates these two regions. If these variables are specialized to be all value q, then this is called the q-matrix (over the Euclidean domain <math>\mathbb{Q}[q]</math>) for the arrangement and much information is contained in its [[Smith normal form]].