Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of twopolynomials (resp. by the discriminant of a polynomial). Geometrically such acurve 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 cusp).State-of-the-art numerical algorithms cannot handle the computation of itstopology. The main challenge is to find numerical criteria that guarantee theexistence and the uniqueness of a singularity inside a given box $B$, whileensuring that $B$ does not contain any closed loop of $\mathcal{C}$. We solvethis problem by providing a square deflation system that can be used to certifynumerically whether $B$ contains a singularity $p$. Then we introduce a numericadaptive separation criterion based on interval arithmetic to ensure that thetopology of $\mathcal C$ in $B$ is homeomorphic to the local topology at $p$.