Solution paths of polynomial homotopies do not turn back as the continuation
parameter *t* increases, due to the regularity of the paths, as discussed
in [Li and Sauer 1987].
Therefore an increment-and fix predictor-corrector
method is appropriate: after each increase of *t*, *t* remains fixed
while correcting the solution by Newton's method.
Figure 3 sketches two possible predictor schemes
in the path tracker.

The clustering of solution paths is avoided by tightening the tolerances of the corrector to enforce quadratic convergence of Newton's method in every step.

Only as , we may have to deal with paths converging to
singular solutions and with paths diverging to infinity.
To this end, several *end games* were proposed by Morgan, Sommese
and Wampler
[1991
,1992a,
1992b]
and by
Sosonkina, Watson and Stewart in [1996] .
Polyhedral end games
[Huber and Verschelde 1998]
provide a certificate of divergence that
allows to separate diverging paths from the rest, without first having to
compute the actual values of the diverging paths accurately.
Next we summerize the idea of [Huber and Verschelde 1998].

A solution path is represented by the following power series expansion:

(7) |

To check whether a solution path really diverges is equivalent to the test on the value for . A first-order approximation of can be computed by

(8) |

A parallel development to make resultants deal with situations when the mixed volume overshoots the number of roots is described in [Rojas 1997].