A baby step-giant step roadmap algorithm for general algebraic sets

Let R be a real closed field and D ⊂ R an ordered domain. We give an algorithm that takes as input a polynomial Q ⊂ D[X1, . . . , Xk], and computes a description of a roadmap of the set of zeros, Zer(Q,R), of Q in R. The complexity of the algorithm, measured by the number of arithmetic operations in the domain D, is bounded by d √ , where d = deg(Q) ≥ 2. As a consequence, there exist algorithms for computing the number of semi-algebraically connected components of a real algebraic set, Zer(Q,R), whose complexity is also bounded by d √ , where d = deg(Q) ≥ 2. The best previously known algorithm for constructing a roadmap of a real algebraic subset of R defined by a polynomial of degree d had complexity d 2).