## 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. [3].

Consider the ideal in (or where denotes the field of rational numbers), formed by taking linear combinations of the coefficients in Eq. (114*), , , , . 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 four 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 four-tuples by multiplying by the matrix to obtain . The row vectors , , of form the set : Thus, we obtain three more generators, which gives us a total of seven generators of the required module of syzygies.