However, it is important to realize that we do not have a vector space here. Three independent constraints on a six-tuple do not produce a space which is necessarily generated by three basis elements. This conclusion would follow if the solutions formed a vector space but they do not. The polynomial six-tuple , can be multiplied by polynomials in , , (scalars) which do not form a field. Thus the inverse in general does not exist within the ring of polynomials. We therefore have a module over the ring of polynomials in the three variables , , . First we present the general methodology for obtaining the solutions to Equation (20) and then apply it to Equation (20).
There are three linear constraints on the polynomials given by Equation (20). Since the equations are linear, the solutions space is a submodule of the module of six-tuples of polynomials. The module of six-tuples is a free module, i.e. it has six basis elements that not only generate the module but are linearly independent. A natural choice of the basis is with 1 in the -th place and 0 everywhere else; runs from 1 to 6. The definitions of generation (spanning) and linear independence are the same as that for vector spaces. A free module is essentially like a vector space. But our interest lies in its submodule which need not be free and need not have just three generators as it would seem if we were dealing with vector spaces.
The problem at hand is of finding the generators of this submodule, i.e. any element of the submodule should be expressible as a linear combination of the generating set. In this way the generators are capable of spanning the full submodule or generating the submodule. In order to achieve our goal, we rewrite Equation (20) explicitly component-wise:
The first step is to use Gaussian elimination to obtain and in terms of ,
We will assume that the polynomials have rational coefficients, i.e. the coefficients belong to , the field of the rational numbers. The set of polynomials form a ring - the polynomial ring in three variables, which we denote by . The polynomial vector . The set of solutions to Equation (27) is just the kernel of the homomorphism , where the polynomial vector is mapped to the polynomial . Thus the solution space is a submodule of . It is called the module of syzygies. The generators of this module can be obtained from standard methods available in the literature. We briefly outline the method given in the books by Becker et al. , and Kreuzer and Robbiano  below. The details have been included in Appendix A.
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