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K-tuple K, and every degeneration T of T , we have ω As before, we must consider several cases. Note that n + 1 ∈ K, and we agree to use the same symbol for the underlying set. Similarly, if L is a set which does not contain n + 1, then we write L for the corresponding tuple in the standard order. The following identity will be useful in several parts. If J ⊂ [n], then ( m∈J ym am )∂aJ = ( m∈J ym )aJ . 48 1 Algebraic Combinatorics Case 1: T ∈ Dep(T )q+1 is a circuit with n + 1 ∈ T Write T = (U, n + 1), and assume that U = (n − q + 1, .

Assuming that m(U1 ,p,n+1) (T ) = 2 leads to a similar argument. We consider the coefficient αn+1 and conclude that (U1 , p) ∈ Dep(T , T ) and hence (U, p) ∈ Dep(T , T ). This contradicts the assumption that all T -relevant sets S belong to a Type III family. 7. Let T ∈ Dep(T )q+1 be a circuit. Suppose T gives rise to codimension-one degenerations of both Type II and Type III. Then the Type II family is unique. For each Type III family there is a unique p ∈ [n + 1] − T so that (Ti , p) is also in the unique Type II family.

Then H(A(A), t) = π(A, t). Proof. If A is empty, then H(A(A), t) = 1 = π(A, t). Let (A, A , A ) be a deletion-restriction triple of a nonempty arrangement. 4 that the Hilbert series satisfies the recursion H(A(A), t) = H(A(A ), t) + tH(A(A ), t). 1). 5 The NBC Complex Definitions This section is borrowed from [39]. 6 to compute the cohomology groups of a seemingly unrelated 28 1 Algebraic Combinatorics complex. 2 to calculate the local system cohomology groups of the complement. Recall the definition of the nbc set from the last section.

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A survey of binary systems by Bruck R.H.

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