We give a separation bound for an isolated multiple root x of a square
multivariate analytic system f satisfying that an operator deduced by adding
Df(x) and a projection of D2f(x) in a direction of the kernel of Df(x)
is invertible. We prove that the deflation process applied on f and this kind
of roots terminates after only one iteration. When x is only given
approximately, we give a numerical criterion for isolating a cluster of zeros
of f near x. We also propose a lower bound of the number of roots in the
cluster.Comment: 17 page