Given a polynomial system f associated with a simple multiple zero x of multiplicity {mu}, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f. If x is only given with limited accuracy, we propose a numerical criterion that f is certified to have {mu} zeros (counting multiplicities) in a small ball around x. Furthermore, for simple double zeros and simple triple zeros whose Jacobian is of normalized form, we define modified Newton iterations and prove the quantified quadratic convergence when the starting point is close to the exact simple multiple zero. For simple multiple zeros of arbitrary multiplicity whose Jacobian matrix may not have a normalized form, we perform unitary transformations and modified Newton iterations, and prove its non-quantified quadratic convergence and its quantified convergence for simple triple zeros.
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