Fast searching in a real algebraic manifold with applications to geometric complexity

Abstract

This paper generalizes the multidimensional searching scheme of Dobkin and Lipton [SIAM J. Comput. 5(2), pp. 181–186, 1976] for the case of arbitrary (as opposed to linear) real algebraic varieties. Let d,r be two positive constants and let P1,…,Pn be n rational r-variate polynomials of degree ≤d. Our main result is an O(n2r + 6) data structure for computing the predicate [∃i (1≤i≤n)|Pi(x)=0] in O(log n) time, for any x∈Er. The method is intimately based on a decomposition technique due to Collins [Proc. 2nd GI Conf. on Automata Theory and Formal Languages, pp. 134–183, 1975]. The algorithm can be used to solve problems in computational geometry via a locus approach. We illustrate this point by deriving an o(n2) algorithm for computing the time at which the convex hull of n (algebraically) moving points in E2 reaches a steady state.