  Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of
two polynomials (resp. by the discriminant of a polynomial). Geometrically such
a curve is the projection of the intersection of the surfaces
$P(x,y,z)=Q(x,y,z)=0$ (resp. $P(x,y,z)=\frac{\partial P}{\partial
z}(x,y,z)=0$), and generically its singularities are nodes (resp. nodes and
ordinary cusps). State-of-the-art numerical algorithms compute the topology of
smooth curves but usually fail to certify the topology of singular ones. The
main challenge is to find practical numerical criteria that guarantee the
existence and the uniqueness of a singularity inside a given box $B$, while
ensuring that $B$ does not contain any closed loop of $\mathcal{C}$. We solve
this problem by first providing a square deflation system, based on
subresultants, that can be used to certify numerically whether $B$ contains a
unique singularity $p$ or not. Then we introduce a numeric adaptive separation
criterion based on interval arithmetic to ensure that the topology of $\mathcal
C$ in $B$ is homeomorphic to the local topology at $p$. Our algorithms are
implemented and experiments show their efficiency compared to state-of-the-art
symbolic or homotopic methods.
