Some Speed-Ups and Speed Limits for Real Algebraic Geometry

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Author: Rojas J.M.

Source: Journal of Complexity, Volume 16, Number 3, September 2000 , pp. 552-571(20)

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Abstract:

We give new positive and negative results, some conditional, on speeding up computational algebraic geometry over the reals: 1. A new and sharper upper bound on the number of connected components of a semi-algebraic set. Our bound is novel in that it is stated in terms of the volumes of certain polytopes and, for a large class of inputs, beats the best previous bounds by a factor exponential in the number of variables. 2. A new algorithm for approximating the real roots of certain sparse polynomial systems. Two features of our algorithm are (a) arithmetic complexity polylogarithmic in the degree of the underlying complex variety (as opposed to the super-linear dependence in earlier algorithms) and (b) a simple and efficient generalization to certain univariate exponential sums. 3. Detecting whether a real algebraic surface (given as the common zero set of some input straight-line programs) is not smooth can be done in polynomial time within the classical Turing model (resp. BSS model over Copf ) only if P=NP (resp. NP sube BPP). The last result follows easily from an unpublished observation of S. Smale. Copyright 2000 Academic Press.