|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 The simplest case of quadratic turning points is well understood,
but these methods no longer work for general types of singularities. We
have experimented with criteria to monitor the Jacobian in order not to
miss any singular solutions along a path. In case of higher order
singularities more accurate predictors are needed, otherwise we do not
get in the range for which reconditioning methods such as deflation can
be applied. Our methods are implemented in the software package PHCpack
and applied to a wide range of polynomial systems arising in various
fields of science and engineering. This is joint work with Kathy Piret.