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.