On valuations, the characteristic polynomial, and complex subspace arrangements
This paper presents a method to determine the characteristic polynomial of a subspace arrangement over an infinite field \({\mathbf k}\), as defined for instance in the book of \textit{P. Orlik} and \textit{H. Terao} [`Arrangements of hyperplanes' (1992; Zbl 0757.55001)]. Let \(V\) be a finite dimensional vector space over \({\mathbf k}\), and \(B\) the Boolean algebra generated by the affine subspaces of \(V\). By an extension theorem of \textit{H. Groemer} [Pac. J. Math. 75, 397-410 (1978; Zbl 0384.52001)], it may be seen that there exists a unique valuation \(\nu:B\rightarrow\mathbb{Z}[t]\) such that \(\nu(A)=t^{\dim(A)}\) for every nonempty affine subspace \(A\) of \(V\). Now the main theorem says that the characteristic polynomial of a subspace arrangement \({\mathcal A}=\{A_1,\ldots,A_k\}\) in \(V\) is given by \(\chi({\mathcal A})=\nu(V-\bigcup_{i=1}^k A_i)\). The computation of the valuation \(\nu\) is facilitated by a Fubini-type theorem. The method is applied to several examples, for instance to a generalization of the so-called Dowling arrangement. As a corollary of the main theorem, the formula of Zaslavsky on the number of chambers of a real hyperplane arrangement is obtained.
scientific article; zbMATH DE number 1146301
scientific article