Pseudozeros of multivariate polynomials

The pseudozero set of a system f of polynomials in n complex variables is the subset of Cn which is the union of the zero-sets of all polynomial systems g that are near to f in a suitable sense. This concept is made precise, and general properties of pseudozero sets are established. In particular it is shown that in many cases of natural interest, the pseudozero set is a semialgebraic set. Also, estimates are given for the size of the projections of pseudozero sets in coordinate directions. Several examples are presented illustrating some of the general theory developed here. Finally, algorithmic ideas are proposed for solving multivariate polynomials.