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By Peter Orlik

An association of hyperplanes is a finite selection of codimension one affine subspaces in a finite dimensional vector house. preparations have emerged independently as very important items in a number of fields of arithmetic corresponding to combinatorics, braids, configuration areas, illustration conception, mirrored image teams, singularity conception, and in laptop technology and physics. This booklet is the 1st complete examine of the topic. It treats preparations with tools from combinatorics, algebra, algebraic geometry, topology, and workforce activities. It emphasizes common recommendations which light up the connections one of the varied facets of the topic. Its major objective is to put the principles of the idea. for this reason, it truly is primarily self-contained and proofs are supplied. however, there are a number of new effects the following. specifically, many theorems that have been formerly recognized just for crucial preparations are proved the following for the 1st time in completegenerality. The textual content presents the complex graduate scholar access right into a very important and lively sector of analysis. The operating mathematician will findthe booklet important as a resource of simple result of the idea, open difficulties, and a entire bibliography of the subject.

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In the Boolean arrangement, every element is modular, so we may take any maximal chain. 3. Thus a maximal chain of modular elements is given by V < {x = O} < {x = y = O} < {O}. Not all elements of the braid arrangement are modular, but V < {Xl = X2} < {Xl = X2 = X3} < ... < {Xl = X2 = ... = Xi} = T is a maximal chain of modular elements. 34 Let A be an arrangement and let L Mobius function /-LA = /-L : L x L --t 7L. as follows: L(A). 2 The Mobius Function JL(X,X) = 1 ~x

By the inductive assumption, we have r(Y) = i - 1. Therefore we have r(Y V Hi) = r(Y) + r(Hi) - r(Y 1\ Hi) (i-l)+l-r(V)=i. This shows that the partition (1ft, ... , 1ft) is independent. Next let X E L \ {V}. Let j be the largest integer such that V = X 1\ X j . ) r(X V Xj) (r(X) + r(Xj ) - r(X 1\ Xj)) 1. This implies that X 1\ X j+! is a hyperplane belonging to A. Thus Ax n 1fj+! = {X 1\ Xj+d is a singleton. 88 that if A has a nice partition 11" = (11"1, ... ) and bi = 11I"il, then 1I"(A, t) = • II (1 + b;t).

Q(8) = xyz(x + y + z)(x + y - z)(x - y + z)(x - Y - z). The reader should check that 71'(A, t) = (1 + t)(1 + 3t)(1 + 3t) = 71'(8, t). However, these arrangements are not L-equivalent. 8 shows that A has two lines which are contained in four hyperplanes. These appear in the picture as the two common points on the line at infinity of the two sets of three parallel lines. 8 also shows that 8 has no such lines. The factorization of these Poincare polynomials is remarkable. Next we prove some general results to explain their factorization.

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