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. [2
], and Kreuzer and Robbiano [16
] below. The details have
been included in Appendix A.
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