syz(Matrix) -- compute the syzygy matrix

Synopsis

• Function: syz

• Usage:

  syz h

• Inputs:
  • h, a matrix, a matrix
• Optional inputs:
  • Algorithm => ..., default value Inhomogeneous, see gb(...,Algorithm=>...)
  • BasisElementLimit => ..., default value infinity, see gb(...,BasisElementLimit=>...)
  • DegreeLimit => ..., default value {}, see gb(...,DegreeLimit=>...)
  • GBDegrees => ..., default value null, see gb(...,GBDegrees=>...)
  • HardDegreeLimit => ..., default value null, see gb(...,HardDegreeLimit=>...)
  • MaxReductionCount => an integer, default value 10, the maximum number of reductions of an S-pair done before requeueing it, if the Inhomogeneous algorithm is in use
  • PairLimit => ..., default value infinity, see gb(...,PairLimit=>...)
  • StopBeforeComputation => ..., default value false, see gb(...,StopBeforeComputation=>...)
  • Strategy => ..., default value {}, see gb(...,Strategy=>...)
  • SyzygyLimit => ..., default value infinity, see gb(...,SyzygyLimit=>...)
  • SyzygyRows => ..., default value infinity, see gb(...,SyzygyRows=>...)
• Outputs:
  • a matrix, the matrix of minimal or trimmed generators for the syzygies among the columns of h

Description

i1 : R = QQ[a..g];

i2 : I = ideal"ab2-c3,abc-cef,ade-cfg"

               2    3
o2 = ideal (a*b  - c , a*b*c - c*e*f, a*d*e - c*f*g)

o2 : Ideal of R

i3 : syz gens I     	       

o3 = {3} | -abc+cef 0        -ade+cfg de2f-bcfg  |
     {3} | ab2-c3   -ade+cfg 0        -c2de+b2fg |
     {3} | 0        abc-cef  ab2-c3   bc3-b2ef   |

             3       4
o3 : Matrix R  <--- R

See also

• gb -- compute a Gröbner basis