syz(Matrix) returns nonminimal syzygies

[aalmousa]

This is related to the example that produced the bug in #3017. In the documentation for syz(Matrix), the output is "the matrix of minimal or trimmed generators for the syzygies among the columns of h". However, this is not true:

i1 : R = QQ[x]

o1 = R

o1 : PolynomialRing

i2 : h = transpose matrix{{x^3+1},{x^2+1}}

o2 = {-3} | x3+1 x2+1 |

             1      2
o2 : Matrix R  <-- R

i3 : syz h

o3 = | x2+1  x3+x2+x+1  |
     | -x3-1 -x4-x3-x-1 |

             2      2
o3 : Matrix R  <-- R

i4 : mingens image oo

o4 = | -x2-1 |
     | x3+1  |

             2      1
o4 : Matrix R  <-- R

I am not sure if the solution here is to fix the code, or to fix the documentation -- the documentation for syz(GroebnerBasis) reiterates that the output might include nonminimal syzygies, but the documentation for syz(Matrix) is misleading.

[mikestillman]

I think we should fix the documentation. In these instances, I'm not sure what the mathematical notion is for minimality. Although, in this case, I think the second column is a polynomial multiple of the first, so there should be some function that could recognize that. One option is during syzygy accumulation, the program could just choose not to append elements which are an obvious multiple of one of the previous ones.