{\bf Speeding Up Real Algebraic Geometry }

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\centerline{J. Maurice Rojas, City University of Hong Kong}

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\noindent         We present a new and sharper upper bound on the number of
connected
components of a semialgebraic set.  Our new 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.
        We then show how to similarly lower earlier complexity
bounds for the existential theory of the reals.

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