  We develop a new symbolic-numeric algorithm for the certification of singular
isolated points, using their associated local ring structure and certified
numerical computations. An improvement of an existing method to compute inverse
systems is presented, which avoids redundant computation and reduces the size
of the intermediate linear systems to solve. We derive a one-step deflation
technique, from the description of the multiplicity structure in terms of
differentials. The deflated system can be used in Newton-based iterative
schemes with quadratic convergence. Starting from a polynomial system and a
small-enough neighborhood, we obtain a criterion for the existence and
uniqueness of a singular root of a given multiplicity structure, applying a
well-chosen symbolic perturbation. Standard verification methods, based eg. on
interval arithmetic and a fixed point theorem, are employed to certify that
there exists a unique perturbed system with a singular root in the domain.
Applications to topological degree computation and to the analysis of real
branches of an implicit curve illustrate the method.
