A New Algorithm to Find a Point in Every Cell Defined by a Family of Polynomials

Abstract

We consider s polynomials P₁,…,P _s in k < s variables with coefficients in an ordered domain A contained in a real closed field 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 P₁,…,P _s . The output is the set of points together with the sign condition at each point. The algorithm uses s(s/k)^kd^{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.