# 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 *P* _{1},...,*P* _{ n } be *n* rational *r*-variate polynomials of degree ≤*d*. Our main result is an \(O(n^{2^{r + 6} } )\) data structure for computing the predicate [∃*i* (1≤*i*≤*n*)|*P* _{ i }(*x*)=0] in *O*(log *n*) time, for any *x*∈*E* ^{ r }. 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*(*n* ^{2}) algorithm for computing the time at which the convex hull of *n* (algebraically) moving points in *E* ^{2} reaches a steady state.

## Keywords

Convex Hull Real Root Rational Coefficient Computational Geometry Binary Search## References

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