Determining dimension of the solution component that contains a computed zero of a polynomial system

When the Jacobian of a computed numerical solution of a polynomial system in Cn allows very small singular values, the solution could be isolated with a multiple multiplicity or may belong to a solution component with positive dimension. The algorithm constructed in this article intends to differentiate those cases by determining the dimension of the solution component M in which the solution lies. Of particular interest is the case when dim(M)=0, then the solution is of course isolated. While the proposed algorithm is experimental, it has been tested successfully on the class of problems with the solution in question belonging to a reduced component. Numerical results are provided to illustrate the accuracy of the algorithm.