Finding connected components of a semialgebraic set in subexponential time

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Let a semialgebraic set be given by a quantifier-free formula of the first-order theory of real closed fields with atomic subformulae of type (f i ≥ 0), 1 ≤i ≤k where the polynomialsf i ε ℤ[X 1,..., Xn] have degrees deg(f i <d and the absolute value of each (integer) coefficient off i is at most 2 M . An algorithm is designed which finds the connected components of the semialgebraic set in time $M^{O(1)} (kd)^{n^{O(1)} } $ . The best previously known bound $M^{O(1)} (kd)^{n^{O(n)} } $ for this problem follows from Collins' method of Cylindrical Algebraic Decomposition.