A Generators of the Module of Syzygies
We require the 4-tuple solutions
![(q3,q′1,q2′,q′3)](article1002x.gif)
to the equation
where for convenience we have substituted
![x = 𝒟1](article1004x.gif)
,
![y = 𝒟2](article1005x.gif)
,
![z = 𝒟3](article1006x.gif)
.
![q3](article1007x.gif)
,
![q1′](article1008x.gif)
,
![q′2](article1009x.gif)
,
![q3′](article1010x.gif)
are
polynomials in
![x](article1011x.gif)
,
![y](article1012x.gif)
,
![z](article1013x.gif)
with integral coefficients, i.e., in
![Z [x, y,z]](article1014x.gif)
.
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
![fi](article1023x.gif)
,
![i = 1,2,3, 4](article1024x.gif)
, and
![gj](article1025x.gif)
,
![j = 1,2,3](article1026x.gif)
, generate the same ideal because we can express
one generating set in terms of the other and vice-versa:
where
![d](article1028x.gif)
and
![c](article1029x.gif)
are
![4 × 3](article1030x.gif)
and
![3 × 4](article1031x.gif)
polynomial matrices, respectively, and are given by
The generators of the 4-tuple module are given by the set
![A ⋃ B ∗](article1033x.gif)
, where
![A](article1034x.gif)
and
![B ∗](article1035x.gif)
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
![𝒢](article1041x.gif)
. The S-polynomial of two polynomials
![g1,g2](article1042x.gif)
is obtained by multiplying
![g1](article1043x.gif)
and
![g2](article1044x.gif)
by suitable terms and then adding, so that the highest terms cancel. For example in our case
![g1 = z2 − 1](article1045x.gif)
and
![g2 = y2 − 1](article1046x.gif)
, and the highest terms are
![z2](article1047x.gif)
for
![g1](article1048x.gif)
and
![y2](article1049x.gif)
for
![g2](article1050x.gif)
. Multiply
![g1](article1051x.gif)
by
![y2](article1052x.gif)
and
![g2](article1053x.gif)
by
![z2](article1054x.gif)
and subtract. Thus, the S-polynomial
![p12](article1055x.gif)
of
![g1](article1056x.gif)
and
![g2](article1057x.gif)
is
Note that order is defined (
![x ≫ y ≫ z](article1059x.gif)
) and the
![y2z2](article1060x.gif)
term cancels. For the Gröbner basis of 3 elements
we get 3 S-polynomials
![p12](article1061x.gif)
,
![p13](article1062x.gif)
,
![p23](article1063x.gif)
. The
![pij](article1064x.gif)
must now be re-expressed in terms of the Gröbner
basis
![𝒢](article1065x.gif)
. This gives a
![3 × 3](article1066x.gif)
matrix
![b](article1067x.gif)
. The final step is to transform to four-tuples by multiplying
![b](article1068x.gif)
by the matrix
![c](article1069x.gif)
to obtain
![∗ b = b ⋅ c](article1070x.gif)
. The row vectors
![∗ bi](article1071x.gif)
,
![i = 1,2,3](article1072x.gif)
, of
![∗ b](article1073x.gif)
form the set
![∗ B](article1074x.gif)
:
Thus, we obtain three more generators, which gives us a total of seven generators of the required module of
syzygies.