  Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total
degree $d$ with integer coefficients of bitsize less than $\tau$, and let
$V_{\mathbb{R}}
  (P) := \{ (x,y) \in \mathbb{R}^2, P (x,y) = 0 \}$ be the real planar
algebraic curve implicitly defined as the vanishing set of $P$. We give a
deterministic and certified algorithm to compute the topology of
$V_{\mathbb{R}} (P)$ in terms of a straight-line planar graph $\mathcal{G}$
that is isotopic to $V_{\mathbb{R}} (P)$. Our analysis yields the upper bound
$\tilde O (d^5 \tau + d^6)$ on the bit complexity of our algorithm, which
matches the current record bound for the problem of computing the topology of a
planar algebraic curve However, compared to existing algorithms with comparable
complexity, our method does not consider any change of coordinates, and the
returned graph $\mathcal{G}$ yields the cylindrical algebraic decomposition
information of the curve.
  Our result is based on two main ingredients: First, we derive amortized
quantitative bounds on the the roots of polynomials with algebraic coefficients
as well as adaptive methods for computing the roots of such polynomials that
actually exploit this amortization. Our second ingredient is a novel approach
for the computation of the local topology of the curve in a neighborhood of all
critical points.
