### 4.5 Generating set for the module of syzygies

The generating set for the module is obtained by further following the procedure in the literature [216]. The details are given in Appendix A, specifically for our case. We obtain 7 generators for the module. These generators do not form a minimal set and there are relations between them; in fact this method does not guarantee a minimum set of generators. These generators can be expressed as linear combinations of , , , and also in terms of , , , given below in Equation (30). The importance in obtaining the 7 generators is that the standard theorems guarantee that these 7 generators do in fact generate the required module. Therefore, from this proven set of generators we can check whether a particular set is in fact a generating set. We present several generating sets below.

Alternatively, we may use a software package called Macaulay 2 which directly calculates the generators given the Equations (25). Using Macaulay 2, we obtain six generators. Again, Macaulay’s algorithm does not yield a minimal set; we can express the last two generators in terms of the first four. Below we list this smaller set of four generators in the order :

Note that the last three generators are just , , . An extra generator is needed to generate all the solutions.

Another set of generators which may be useful for further work is a Gröbner basis of a module. The concept of a Gröbner basis of an ideal can be extended to that of a Gröbner basis of a submodule of where is a field, since a module over the polynomial ring can be considered as generalization of an ideal in a polynomial ring. Just as in the case of an ideal, a Gröbner basis for a module is a generating set with special properties. For the module under consideration we obtain a Gröbner basis using Macaulay 2:

Note that in this Gröbner basis , , , . Only is the new generator.

Another set of generators are just , , , and . This can be checked using Macaulay 2, or one can relate , , , and to the generators , , by polynomial matrices. In Appendix B, we express the 7 generators we obtained following the literature, in terms of , , , and . Also we express , , , and in terms of . This proves that all these sets generate the required module of syzygies.

The question now arises as to which set of generators we should choose which facilitates further analysis. The analysis is simplified if we choose a smaller number of generators. Also we would prefer low degree polynomials to appear in the generators so as to avoid cancellation of leading terms in the polynomials. By these two criteria we may choose or , , , . However, , , , possess the additional property that this set is left invariant under a cyclic permutation of indices . It is found that this set is more convenient to use because of this symmetry.