(joint with Jan Verschelde (University of Illinois at Chicago))
To compute critical points of polynomial systems with parameters we may
apply the Jacobian criterion. A major shortcoming of this Jacobian
criterion is that the augmented system may get too large to solve.
Therefore, we locally apply a so-called sweep for critical values,
tracking solution paths for a range of the parameter values.
Our applications include polynomial systems arising in models of neural
networks, molecular configurations and symmetrical Stewart-Gough platforms.
We have satisfactory experience locating quadratic turning points.
Currently we are investigating the use of higher-order derivatives
to detect and compute higher-order critical points via a sweep.