, Volume 2, Issue 2, pp 133-186

Counting connected components of a semialgebraic set in subexponential time

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Abstract

Let a semialgebraic set be given by a quantifier-free formula of the first-order theory of real closed fields withk atomic subformulae of the typef i≥0 for 1≤ik, where the polynomialsf i∈ℤ[X 1,...,X n] have degrees deg(f i)<d and the absolute value of each (integer) coefficient off i is at most 2M. An algorithm is exhibited which counts the number of connected components of the semialgebraic set in time (M (kd)n 20)O (1). Moreover, the algorithm allows us to determine whether any pair of points from the set are situated in the same connected component.