  We propose a symbolic-numeric algorithm to count the number of solutions of a
polynomial system within a local region. More specifically, given a
zero-dimensional system $f_1=\cdots=f_n=0$, with
$f_i\in\mathbb{C}[x_1,\ldots,x_n]$, and a polydisc
$\mathbf{\Delta}\subset\mathbb{C}^n$, our method aims to certify the existence
of $k$ solutions (counted with multiplicity) within the polydisc.
  In case of success, it yields the correct result under guarantee. Otherwise,
no information is given. However, we show that our algorithm always succeeds if
$\mathbf{\Delta}$ is sufficiently small and well-isolating for a $k$-fold
solution $\mathbf{z}$ of the system.
  Our analysis of the algorithm further yields a bound on the size of the
polydisc for which our algorithm succeeds under guarantee. This bound depends
on local parameters such as the size and multiplicity of $\mathbf{z}$ as well
as the distances between $\mathbf{z}$ and all other solutions. Efficiency of
our method stems from the fact that we reduce the problem of counting the roots
in $\mathbf{\Delta}$ of the original system to the problem of solving a
truncated system of degree $k$. In particular, if the multiplicity $k$ of
$\mathbf{z}$ is small compared to the total degrees of the polynomials $f_i$,
our method considerably improves upon known complete and certified methods.
  For the special case of a bivariate system, we report on an implementation of
our algorithm, and show experimentally that our algorithm leads to a
significant improvement, when integrated as inclusion predicate into an
elimination method.
