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# syz(GroebnerBasis) -- retrieve the syzygy matrix

## Synopsis

• Function: syz
• Usage:
syz G
• Inputs:
• G, , 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:
• , 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