On the number of Bounding Cycles for Nonlinear Arrangements

For a real hyperplane arrangement A ⊂ R, among the first invariants that were determined for A were the number of chambers in the complement Rn\A by Zavslavsky [Za] and the number of bounded chambers by Crapo [Cr]. In the consideration of certain classes of hypergeometric functions, there also arise arrangements of hypersurfaces which need not be hyperplanes (see e.g. Aomoto [Ao]). In this paper we will obtain a formula for the number of bounded regions (i.e. chambers) in the complement of a nonlinear arrangement of hypersurfaces. For example, for the general position arrangements of quadrics in Figure 1, we see the number of bounded regions in the complement are respectively 1, 5, and 13.