Singular

i got nothing about this annihiliator

why?

if you got result, could you post your code

Well, I tried this in both Singular and Macaulay2. I got a non-trivial answer in Macaulay2 given the matrix (or module) as written, but needed to transpose it to get a similar result in Singular.

in the book introduction to groebner bases
page 178

i input the following, syz is only 0,

it said using degrevlex , x>y>z

ring r = 0,(x,y,z),dp

module MD = [1,x2+y,x*y-y*z,0,0],[0,x*z-y,z-x,0,0],[0,0,0,x2+y,x*y-y*z],[1,0,0,x*z-y,z-x];

module MK = syz(MD);

matrix MM = MK;

print(MM);

module N =[x2+y,x*z-y],[x*y-y*z,z-x];

quotient(N,freemodule(nrows(MK)));