Title: Computing Simple Multiple Zeros of Polynomial Systems,
Abstract: 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.