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

Abstract. 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, Rk ), of Q in Rk . The complexity of the algorithm, measured by the number of arithmetic operations in the domain D, is bounded by dO(k √ k) , 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, Rk ), whose complexity is also bounded by dO(k √ k) , where d = deg(Q) ≥ 2. The best previously known algorithm for constructing a roadmap of a real algebraic subset of Rk defined by a polynomial of degree d had complexity dO(k2 ) .

• Type: Journal
• Year: 2012
• Where: CORR
• Authors: Saugata Basu, Marie-Françoise Roy, Mohab Safey El Din, Éric Schost
• Keywords: Arithmetic Operations, CORR 2012, Education, Ordered Domain, Xk