A New Algorithm to Find a Point in Every

We consider s polynomials P 1 ; : : :; P s in k < s variables with coef-cients in an ordered domain A contained in a real closed eld R, each of degree at most d. We present a new algorithm which computes a point in each connected component of each non-empty sign condition over P 1 ; : : :; P s. The output is the set of points together with the sign condition at each point. The algorithm uses s(s=k) k d O(k) arithmetic operations in A. The algorithm is nearly optimal in the sense that the size of the output can be as large as s(O(sd=k)) k. Previous algorithms of Canny and Renegar used (sd) O(k) operations 5, 7, 8, 15]. We use either these algorithms in the case s = 1 as a subroutine in our algorithm. As a bonus, our algorithm yields an independent proof of the bound on the number of connected components in all non-empty sign conditions ((14]) and also yields an independent proof of a theorem of Warren 1 (see 17]) which is used in 14] as well as in many other problems of a combinatorial nature (see Alon 1] for example).