This is a survey talk about some recent symbolic-numeric techniques to solve ill-conditioned multivariate polynomial systems. In particular, we will concentrate on systems that are over-constrained or have roots with multiplicities, and are given with inexact coefficients. First I give some theoretical background on polynomial systems with inexact coefficients, ill-posed and ill-conditioned problems, and on the objectives when trying to solve these systems. Next, I will describe a family of iterative techniques which, for a given inexact system of polynomials and given root structure, computes the nearest system which has roots with the given structure. Finally, I present a global method to solve multivariate polynomial systems which are near root multiplicities and thus have clusters of roots. The method computes a new system which is "square-free", i.e. it has exactly one root in each cluster near the arithmetic mean of the cluster. This method is global in the sense that it works simultaneously for all clusters. The results presented in the talk are joint work with Itnuit Janovitz-Freireich, Bernard Mourrain, Scott Pope, Lajos Ronyai, Olivier Ruatta, and Mark Sciabica.
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