12,175 research outputs found
Some Speed-Ups and Speed Limits for Real Algebraic Geometry
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 semialgebraic 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 C) only if P=NP (resp. NP<=BPP). The last result follows easily
from an unpublished result of Steve Smale.Comment: This is the final journal version which will appear in Journal of
Complexity. More typos are corrected, and a new section is added where the
bounds here are compared to an earlier result of Benedetti, Loeser, and
Risler. The LaTeX source needs the ajour.cls macro file to compil
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