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); ... calling constructEmbedding ... calling phc -c < /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44733-0/2PHCbatch > /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44733-0/3PHCsession output of phc -c is in file /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44733-0/1PHCoutput ... calling solveSystem ... using temporary files /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44733-0/4PHCinput and /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44733-0/4PHCoutput ... constructing a witness set ... |
i4 : w o4 = [dim=1,deg=4] o4 : WitnessSet |
i5 : L = factorWitnessSet(w) preparing input file to /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44733-0/5PHCinput preparing batch file to /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44733-0/7PHCbatch ... calling monodromy breakup ... session information of phc -f is in /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44733-0/8PHCsession output of phc -f is in file /var/folders/40/dy88l5qd361391m_3v2m51bm0000gn/T/M2-44733-0/6PHCoutput found 2 irreducible factors o5 = A variety of dimension 1 with components in dim 1: (dim=1,deg=3) (dim=1,deg=1) o5 : NumericalVariety |