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 $D^2f(x)$ 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.

翻译：我们给出了一个关于平方多变量解析系统$f$的孤立多重根$x$的分离界，满足由添加$Df(x)$与$D^2f(x)$在$Df(x)$核方向上的投影所导出的算子是可逆的。我们证明了应用于$f$及此类根的解缩过程仅需一次迭代即可终止。当$x$仅近似给定时，我们提出了一个用于孤立$x$附近$f$的零点簇的数值判据。此外，我们还给出了该簇中根数目的下界。