Divide and Conquer Roadmap for Algebraic Sets

Let $$\mathrm{R}$$R be a real closed field and $$\hbox {D}\subset \mathrm{R}$$D⊂R an ordered domain. We describe an algorithm that given as input a polynomial $$P \in \hbox {D}[ X_{1} , \ldots ,X_{{ k}} ]$$P∈D[X1,…,Xk] and a finite set, $$\mathcal {A}= \{ p_{1} , \ldots ,p_{m} \}$$A={p1,…,pm}, of points contained in $$V= {\mathrm{{Zer}}} ( P, \mathrm{R}^{{ k}})$$V=Zer(P,Rk) described by real univariate representations, computes a roadmap of $$V$$V containing $$\mathcal {A}$$A. The complexity of the algorithm, measured by the number of arithmetic operations in $$\hbox {D}$$D, is bounded by $$\big ( \sum _{i=1}^{m} D^{O ( \log ^{2} ( k ) )}_{i} +1 \big ) ( k^{\log ( k )} d )^{O ( k\log ^{2} ( k ))}$$(∑i=1mDiO(log2(k))+1)(klog(k)d)O(klog2(k)), where $$d= \deg ( P )$$d=deg(P) and $$D_{i}$$Di is the degree of the real univariate representation describing the point $$p_{i}$$pi. The best previous algorithm for this problem had complexity card $$( \mathcal {A} )^{O ( 1 )} d^{O ( k^{3/2} )}$$(A)O(1)dO(k3/2) (Basu et al., ArXiv, 2012), where it is assumed that the degrees of the polynomials appearing in the representations of the points in $$\mathcal {A}$$A are bounded by $$d^{O ( k )}$$dO(k). As an application of our result we prove that for any real algebraic subset $$V$$V of $$\mathbb {R}^{k}$$Rk defined by a polynomial of degree $$d$$d, any connected component $$C$$C of $$V$$V contained in the unit ball, and any two points of $$C$$C, there exists a semi-algebraic path connecting them in $$C$$C, of length at most $$( k ^{\log (k )} d )^{O ( k\log ( k ) )}$$(klog(k)d)O(klog(k)), consisting of at most $$( k ^{\log (k )} d )^{O ( k\log ( k ) )}$$(klog(k)d)O(klog(k)) curve segments of degrees bounded by $$( k ^{\log ( k )} d )^{O ( k \log ( k) )}$$(klog(k)d)O(klog(k)). While it was known previously, by a result of D’Acunto and Kurdyka (Bull Lond Math Soc 38(6):951–965, 2006), that there always exists a path of length $$( O ( d ) )^{k-1}$$(O(d))k-1 connecting two such points, there was no upper bound on the complexity of such a path.