Sweeping Algebraic Curves for Singular Solutions
For researchers solving polynomial systems with parameters, this work provides a method to detect general singular solutions, though it is incremental as it extends existing deflation techniques.
The paper addresses the problem of tracking singular solutions in polynomial systems with parameters, where standard predictor-corrector methods fail for general singularities. It proposes monitoring the determinant of the Jacobian matrix to detect singular points and relates deflation effectiveness to the winding number, with experiments on various applications.
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.