Go to previous page Go up Go to next page

A Generators of the Module of Syzygies

We require the 4-tuple solutions (q3,q'1,q'2,q'3) to the equation
(1- xyz) q + (xz - y)q' + x(1 - z2)q' + (1 - x2)q' = 0, (81) 3 1 2 3
where for convenience we have substituted x = D1, y = D2, z = D3. q3, q'1, q'2, q'3 are polynomials in x, y, z with integral coefficients, i.e. in Z[x, y,z].

We now follow the procedure in the book by Becker et al. [2].

Consider the ideal in Z[x,y,z] (or Q[x, y,z] where Q denotes the field of rational numbers), formed by taking linear combinations of the coefficients in Equation (81View Equation), f1 = 1 - xyz, f2 = xz - y, f3 = x(1 - z2), f4 = 1- x2. A Gröbner basis for this ideal is

2 2 G = {g1 = z - 1,g2 = y - 1,g3 = x - yz}. (82)
The above Gröbner basis is obtained using the function GroebnerBasis in Mathematica. One can check that both the fi, i = 1,2,3, 4, and gj, j = 1,2,3, generate the same ideal because we can express one generating set in terms of the other and vice-versa:
fi = dijgj, gj = cjifi, (83)
where d and c are 4 × 3 and 3 × 4 polynomial matrices, respectively, and are given by
( - 1 - z2 - yz ) ( 2 ) y 0 z 0 0 - x z - 1 d = , c = - 1 -y 0 0 . (84) - x 0 2 0 0 z 1 0 - 1 - z - (x + yz)
The generators of the 4-tuple module are given by the set A U B*, where A and B* are the sets described below:

A is the set of row vectors of the matrix I- d .c where the dot denotes the matrix product and I is the identity matrix, 4× 4 in our case. Thus,

a1 = (z2 - 1,0,x - yz,1- z2), 2 2 a2 = (0,z (1 - z ),xy - z,y (1 - z )), (85) a3 = (0,0, 1- x2,x(z2 - 1)), 2 2 a4 = (- z ,xz, yz,z ).
We thus first get 4 generators. The additional generators are obtained by computing the S-polynomials of the Gröbner basis G. The S-polynomial of two polynomials g1,g2 is obtained by multiplying g1 and g2 by suitable terms and then adding, so that the highest terms cancel. For example in our case g1 = z2 - 1 and g2 = y2- 1, and the highest terms are z2 for g1 and y2 for g2. Multiply g1 by y2 and g2 by z2 and subtract. Thus, the S-polynomial p12 of g1 and g2 is
p = y2g - z2g = z2 - y2. (86) 12 1 2
Note that order is defined (x » y » z) and the y2z2 term cancels. For the Gröbner basis of 3 elements we get 3 S-polynomials p12, p13, p23. The pij must now be re-expressed in terms of the Gröbner basis G. This gives a 3 × 3 matrix b. The final step is to transform to 4-tuples by multiplying b by the matrix c to obtain * b = b .c. The row vectors * bi, i = 1,2,3, of * b form the set * B:
b*1 = (z2- 1,y (z2- 1),x (1- y2),(y2- 1)(z2- 1)), * 2 2 2 b2 = (0,z(1 - z ),1 - z - x (x- yz),(x - yz) (z - 1)) , (87) b*3 = (-x + yz, z- xy,1 - y2,0).
Thus we obtain 3 more generators which gives us a total of 7 generators of the required module of syzygies.
  Go to previous page Go up Go to next page