Probing a Set of Hyperplanes by Lines and Related Problems

Suppose that for a set H of n unknown hyperplanes in the Euclidean d-dimensional space, a line probe is available which reports the set of intersection points of a query line with the hyperplanes. Under this model, this paper investigates the complexity to find a generic line for H and further to determine the hyperplanes in H . This problem arises in factoring the u-resultant to solve systems of polynomials (e.g., Renegar [12]). We prove that d+1 line probes are sufficient to determine H . Algorithmically, the time complexity to find a generic line and reconstruct H from O(dn) probed points of intersection is important. It is shown that a generic line can be computed in O(dn log n) time after d line probes, and by an additional d line probes, all the hyperplanes in H are reconstructed in O(dn log n) time. This result can be extended to the d-dimensional complex space. Also, concerning the factorization of the u-resultant using the partial derivatives on a generic line, we touch upon reducing the time complexity to compute the partial derivatives of the u-resultant represented as the determinant of a matrix