syz(GroebnerBasis) -- retrieve the syzygy matrix

Synopsis

• Function: syz

• Usage:

  syz G

• Inputs:
  • G, a Gröbner basis, the Gröbner basis of a matrix h
• Optional inputs:
  • Algorithm => ..., default value Inhomogeneous, compute the syzygy matrix
  • BasisElementLimit => ..., default value infinity, compute the syzygy matrix
  • DegreeLimit => ..., default value {}, compute the syzygy matrix
  • GBDegrees => ..., default value null, compute the syzygy matrix
  • HardDegreeLimit => ..., default value null, compute the syzygy matrix
  • MaxReductionCount => ..., default value 10, compute the syzygy matrix
  • PairLimit => ..., default value infinity, compute the syzygy matrix
  • StopBeforeComputation => ..., default value false, compute the syzygy matrix
  • Strategy => ..., default value {}, compute the syzygy matrix
  • SyzygyLimit => ..., default value infinity, compute the syzygy matrix
  • SyzygyRows => ..., default value infinity, compute the syzygy matrix
• Outputs:
  • a matrix, the matrix of syzygies among the columns of h

Description

Warning: the result may be zero if syzygies were not to be retained during the calculation, or if the computation was not continued to a high enough degree.

The matrix of syzygies is returned without removing non-minimal syzygies.

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

i2 : I = ideal"ab2-c3,abc-def,ade-bfg"

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

o2 : Ideal of R

i3 : G = gb(I, Syzygies=>true);

i4 : syz G

o4 = {3} | -abc+def 0        -ade+bfg -d2e2f+b2cfg |
     {3} | ab2-c3   -ade+bfg 0        c3de-b3fg    |
     {3} | 0        abc-def  ab2-c3   -bc4+b2def   |

             3      4
o4 : Matrix R  <-- R

i5 : syz gens I

o5 = {3} | -abc+def 0        -ade+bfg |
     {3} | ab2-c3   -ade+bfg 0        |
     {3} | 0        abc-def  ab2-c3   |

             3      3
o5 : Matrix R  <-- R

i6 : mingens image syz G

o6 = {3} | -abc+def 0        -ade+bfg |
     {3} | ab2-c3   -ade+bfg 0        |
     {3} | 0        abc-def  ab2-c3   |

             3      3
o6 : Matrix R  <-- R

See also

• gb -- compute a Gröbner basis
• mingens -- minimal generator matrix