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

TL;DR

This paper introduces a symbolic-numeric method for refining approximate isolated singular solutions of polynomial systems with Jacobian corank one, utilizing regularized Newton iteration and Max Noether conditions for quadratic convergence.

Contribution

The paper presents a novel approach combining regularized Newton iteration and Max Noether conditions to efficiently refine singular solutions with corank one.

Findings

Quadratic convergence of the proposed method near the exact solution

Bounded matrix size of n x n in the algorithm

Effective refinement of approximate singular solutions

Abstract

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.