Divide and Conquer Roadmap for Algebraic Sets

Let R be a real closed field and D subset of R an ordered domain. We describe an algorithm that given as input a polynomial P is an element of D[X-1, ... , X-k] and a finite set, A = {p(1), ... , p(m)}, of points contained in V = Zer(P, R-k) described by real univariate representations, computes a roadmap of V containing A. The complexity of the algorithm, measured by the number of arithmetic operations in D, is bounded by(Sigma D-m(i-1)i(O(log2(k))) + 1 (k(log(k)) d)(O(log2(k))), where d = deg(P) and D-i is the degree of the real univariate representation describing the point p(i). The best previous algorithm for this problem had complexity card (A)(O(1))d(O(k3/2)) (Basu et al., ArXiv, 2012), where it is assumed that the degrees of the polynomials appearing in the representations of the points in A are bounded by d(O(k)). As an application of our result we prove that for any real algebraic subset V of R-k defined by a polynomial of degree d, any connected component C of V contained in the unit ball, and any two points of C, there exists a semi-algebraic path connecting them in C, of length at most (k(log(k)) d)(O(k log(k))), consisting of at most (k(log(k))d)(O(k log(k))) curve segments of degrees bounded by (k(log(k)) d)(O(k log(k))). While it was known previously, by a result of D'Acunto and Kurdyka (Bull Lond Math Soc 38(6): 951-965, 2006), that there always exists a path of length (O(d))(k-1) connecting two such points, there was no upper bound on the complexity of such a path.