Computing Isolated Singular Solutions of Polynomial Systems: Case of Breadth One

We present a symbolic-numeric method to refine an approximate isolated singular solution $\hat{\mathbf{x}}=(\hat{x}_{1}, ..., \hat{x}_{n})$ of a polynomial system $F=\{f_1, ..., f_n\}$ when the Jacobian matrix of $F$ evaluated at $\hat{\mathbf{x}}$ has corank one approximately. Our new approach is based on the regularized Newton iteration and the computation of approximate Max Noether conditions satisfied at the approximate singular solution. The size of matrices involved in our algorithm is bounded by $n \times n$. The algorithm converges quadratically if $\hat{\xx}$ is close to the isolated exact singular solution.