A Generators of the Module of Syzygies

We require the 4-tuple solutions to the equation
where for convenience we have substituted , , . , , , are polynomials in , , with integral coefficients, i.e. in .

We now follow the procedure in the book by Becker et al. [2].

Consider the ideal in (or where denotes the field of rational numbers), formed by taking linear combinations of the coefficients in Equation (81), , , , . A Gröbner basis for this ideal is

The above Gröbner basis is obtained using the function GroebnerBasis in Mathematica. One can check that both the , , and , , generate the same ideal because we can express one generating set in terms of the other and vice-versa:
where and are and polynomial matrices, respectively, and are given by
The generators of the 4-tuple module are given by the set , where and are the sets described below:

is the set of row vectors of the matrix where the dot denotes the matrix product and is the identity matrix, in our case. Thus,

We thus first get 4 generators. The additional generators are obtained by computing the S-polynomials of the Gröbner basis . The S-polynomial of two polynomials is obtained by multiplying and by suitable terms and then adding, so that the highest terms cancel. For example in our case and , and the highest terms are for and for . Multiply by and by and subtract. Thus, the S-polynomial of and is
Note that order is defined () and the term cancels. For the Gröbner basis of 3 elements we get 3 S-polynomials , , . The must now be re-expressed in terms of the Gröbner basis . This gives a matrix . The final step is to transform to 4-tuples by multiplying by the matrix to obtain . The row vectors , , of form the set :
Thus we obtain 3 more generators which gives us a total of 7 generators of the required module of syzygies.