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 componentMin 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. © 2007 Elsevier Inc. All rights reserved.