Finding connected components of a semialgebraic set in subexponential time

AbstractLet a semialgebraic set be given by a quantifier-free formula of the first-order theory of real closed fields with atomic subformulae of type (fi ≥ 0), 1 ≤i ≤k where the polynomialsfi ε ℤ[X1,..., Xn] have degrees deg(fi <d and the absolute value of each (integer) coefficient offi is at most 2M. 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.