A note on the Poincaré polynomial of an arrangement (Q1382784)

The paper is concerned with combinatorial invariants of an arrangement of hyperplanes in real \(\ell\)-space. The main result is a formula expressing the Poincaré-polynomial of the deconing of a central arrangement as a sum over all subspaces \(X\) containing a fixed hyperplane, where each summand factors into some power of the variable, the Poincaré-polynomial of the restriction of the arrangement to \(X\) and the \(\beta\)-invariant of the localization at \(X\). The main result is applied to give alternative proofs for results of \textit{A. Varchenko} [Math. USSR, Izv. 35, No. 3, 543-571 (1990; Zbl 0709.33001)].