factorWitnessSet -- applies monodromy to factor a witness set into irreducible components

Synopsis

Usage:

factorWitnessSet(w)
Inputs:
- w, a witness set, properly embedded with slack variables
Optional inputs:
- Verbose => ..., default value false, option to specify whether additional output is wanted
Outputs:
- a list, a list of witness sets, every element of the list is irreducible
Consequences:
- writes the system to temporary files,
- invokes the command phc -f (with option 2),
- uses monodromy to factor,
- uses default settings of path trackers ,
- stores output of phc in temporary file,
- parses and outputs the solutions.

A witness set is irreducible if there exists a path between any two of its generic points that does not pass through a singularity.

We illustrate the factorization via the twisted cubic and a line.

i1 : R = CC[x,y,z]; f = {(x^2-y)*(x-1), x^3 - z};

i3 : (w,ns) = topWitnessSet(f,1);

i4 : w

o4 = w

o4 : WitnessSet

i5 : L = factorWitnessSet(w)
found 2 irreducible factors 

o5 = L

o5 : NumericalVariety