Sweeping Algebraic Curves for Singular Solutions

TL;DR

This paper introduces a method for detecting singular solutions in polynomial systems by monitoring the Jacobian determinant, enhancing solution path tracking especially at complex singularities.

Contribution

It proposes a novel approach to identify singular solutions in polynomial systems using Jacobian determinant monitoring, extending beyond quadratic turning points.

Findings

Effective detection of singular solutions demonstrated

Relation between deflation effectiveness and winding number established

Computational experiments validate the approach across various applications

Abstract

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. In order not to miss any singular solutions along a path we propose to monitor the determinant of the Jacobian matrix. We examine the operation range of deflation and relate the effectiveness of deflation to the winding number. Computational experiments on systems coming from different application fields are presented.