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

## Synopsis

• Usage:
factorWitnessSet(w)
• Inputs:
• w, , 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.

## Description

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