  In this paper we provide a new method to certify that a nearby polynomial
system has a singular isolated root with a prescribed multiplicity structure.
More precisely, given a polynomial system f $=(f\_1, \ldots, f\_N)\in C[x\_1,
\ldots, x\_n]^N$, we present a Newton iteration on an extended deflated system
that locally converges, under regularity conditions, to a small deformation of
$f$ such that this deformed system has an exact singular root. The iteration
simultaneously converges to the coordinates of the singular root and the
coefficients of the so called inverse system that describes the multiplicity
structure at the root. We use $$\alpha$$-theory test to certify the quadratic
convergence, and togive bounds on the size of the deformation and on the
approximation error. The approach relies on an analysis of the punctual Hilbert
scheme, for which we provide a new description. We show in particular that some
of its strata can be rationally parametrized and exploit these parametrizations
in the certification. We show in numerical experimentation how the approximate
inverse system can be computed as a starting point of the Newton iterations and
the fast numerical convergence to the singular root with its multiplicity
structure, certified by our criteria.
