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

Let $\mathrm{R}$ be a real closed field and $\mathrm{D}\subset \mathrm{R}$ an ordered domain. We give an algorithm that takes as input a polynomial $Q \in $\mathrm{D}$[X_1,\ldots,X_k]$, and computes a description of a roadmap of the set of zeros, $\mathrm{Z}\mathrm{e}\mathrm{r}(Q,{\mathrm{R}}^k)$, of $Q$ in ${\mathrm{R}}^k$. The complexity of the algorithm, measured by the number of arithmetic operations in the ordered domain $\mathrm{D}$, is bounded by $d^{O(k\sqrt{k})}$, where $d = $\mathrm{d}\mathrm{e}\mathrm{g}$(Q)\ge 2$. As a consequence, there exist algorithms for computing the number of semi-algebraically connected components of a real algebraic set, $\mathrm{Z}\mathrm{e}\mathrm{r}(Q,{\mathrm{R}}^k)$, whose complexity is also bounded by $d^{O(k \sqrt{k})}$,where$d = $\mathrm{d}\mathrm{e}\mathrm{g}$(Q)\ge 2$. The best previously known algorithm for constructing a roadmap of a real algebraic subset of ${\mathrm{R}}^k$ defined by a polynomial of degree $d$ has complexity $d^{O(k^2)}$.