# Probing a set of hyperplanes by lines and related problems

## Abstract

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 [13]). 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.

## Keywords

Grid Point Time Complexity Generic Direction Generic Point Dual Space## Preview

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