We analyze the bit complexity of an algorithm for the computation of at least one point in each connected component of a smooth real algebraic set. This work is a continuation of our analysis of the hypersurface case (On the bit complexity of finding points in connected components of a smooth real hypersurface, ISSAC'20). In this paper, we extend the analysis to more general cases. Let $F=(f_1,..., f_p)$ in $\mathbb{Z}[X_1, ... , X_n]^p$ be a sequence of polynomials with $V = V(F) \subset \mathbb{C}^n$ a smooth and equidimensional variety and $\langle F \rangle \subset \mathbb{C}[X_1, ..., X_n]$ a radical ideal. To compute at least one point in each connected component of $V \cap \mathbb{R}^n$, our starting point is an algorithm by Safey El Din and Schost (Polar varieties and computation of one point in each connected component of a smooth real algebraic set, ISSAC'03). This algorithm uses random changes of variables that are proven to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model; here, we analyze the bit complexity and the error probability, and we provide a quantitative analysis of the genericity statements. In particular, we are led to use Lagrange systems to describe polar varieties, as they make it simpler to rely on techniques such as weak transversality and an effective Nullstellensatz.

翻译：我们分析一个算法的位数复杂性, 以计算平滑真实的代数集中每个连接组件中至少一个点。 这项工作是我们对超表层案例的分析的继续( 在平滑的超表层中, ISASAC'20 的连接组件中发现点的比小复杂点 )。 在本文中, 我们将分析扩大到更普通的案例 。 $F= (f_ 1,...) f_ p), 以$\mathbb{[X_ 1,..., X_ n] $ p$ 计算至少一个连接组件的 $V = V( F) = subssset = mathb{C\\\\ $n$ 平滑和 equithexual exmissual case 。 我们的开始点是Safeality El Din 和 Schost( Postality) 的算算法 。 在Silental liveralalalal assal assal assalationsessional coal coal coal coal coal coals reals 中, 我们的计算中, 需要至少要用一个固定的精确到一定的成本和计算。