  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.
