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# syz(GroebnerBasis) -- retrieve the syzygy matrix

- 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

There appear to be 4 syzygies, but the last one is a combination of the first three:

    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

## Ways to use this method:

- syz(GroebnerBasis) -- retrieve the syzygy matrix

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/functions/syz-doc.m2:41:0.