• DOI:10.1007/s10208-014-9212-1
• Corpus ID: 9427498

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

@article{Basu2014ABS,
  title={A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets},
  author={Saugata Basu and Marie-Françoise Roy and Mohab Safey El Din and {\'E}ric Schost},
  journal={Foundations of Computational Mathematics},
  year={2014},
  volume={14},
  pages={1117-1172}
}

• S. Basu, Marie-Françoise Roy, +1 author É. Schost
• Published 31 January 2012
• Mathematics
• Foundations of Computational Mathematics

Let $$\mathrm {R}$$R be a real closed field and $$\mathrm{D}\subset \mathrm {R}$$D⊂R an ordered domain. We present an algorithm that takes as input a polynomial $$Q \in \mathrm{D}[X_{1},\ldots ,X_{k}]$$Q∈D[X1,…,Xk] and computes a description of a roadmap of the set of zeros, $$\mathrm{Zer}(Q,\,\mathrm {R}^{k}),$$Zer(Q,Rk), of Q in $$\mathrm {R}^{k}.$$Rk. The complexity of the algorithm, measured by the number of arithmetic operations in the ordered domain $$\mathrm{D},$$D, is bounded by $$d^{O(k \sqrt{k})},$$dO(kk), where $$d = \deg (Q)\ge 2.$$d=deg(Q)≥2. As a consequence, there exist algorithms for computing the number of semialgebraically connected components of a real algebraic set, $$\mathrm{Zer}(Q,\,\mathrm {R}^{k}),$$Zer(Q,Rk), whose complexity is also bounded by $$d^{O(k \sqrt{k})},$$dO(kk), where $$d = \deg (Q)\ge 2.$$d=deg(Q)≥2. The best previously known algorithm for constructing a roadmap of a real algebraic subset of $$\mathrm {R}^{k}$$Rk defined by a polynomial of degree d has complexity $$d^{O(k^{2})}.$$dO(k2).