  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.
