A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets

Abstract

Let $$\mathrm {R}$$ be a real closed field and $$\mathrm{D}\subset \mathrm {R}$$ an ordered domain. We present 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, $$\mathrm{Zer}(Q,\,\mathrm {R}^{k}),$$ 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 $$d = \deg (Q)\ge 2.$$ As a consequence, there exist algorithms for computing the number of semialgebraically connected components of a real algebraic set, $$\mathrm{Zer}(Q,\,\mathrm {R}^{k}),$$ whose complexity is also bounded by $$d^{O(k \sqrt{k})},$$ where $$d = \deg (Q)\ge 2.$$ 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 $$d^{O(k^{2})}.$$