Betti numbers of modules over Noetherian rings with ... - IPM

Appendix A: Algorithms for our criterion for linear resolutions 93
• Switch to a sparse upper triangular bi-change of coordinates
Note that if | N | is large enough or more precisely when |N|
|B|
is almost 1, we are
advised to do the change ordering. Of course, this is better recognized by experience
since it is quite dependent on the degree of elements of P, the ideal associated to
Rees ring of I and its initial ideal, in(P ). That is if the large powers of P are more
concentrating on t’s than x’s, it is a good idea to choose the term order t < x. But
if it didn’t help yet (which is true most of the time), we use the trick mentioned
in Theorem 2.1.5. That is start the whole story for g(P ) instead of P, where g is
a bi-homogenous isomorphism on K[x, t]. One could try to take g ”generic”, as in
(A.2).
g := g1 × g2,
g1 := xi ↦−→ Random(Sum(x1, · · · , x6)),
g2 := tj ↦−→ Random(Sum(t1, · · · , t10)),
(A.2)
for all i = 1, · · · , 6 and all j = 1, · · · , 10, where by Random(Sum(x1, · · · , x6)) we
mean a linear combination of x1, · · · , x6 with random coefficients and the same inter-
pretation for t1, · · · , t10. But we realized that a properly chosen sparse random upper
triangular g does the job as well. Hence we suggest to use the following algorithm
to generate a sparse random upper triangular bi-change of coordinates. We
choose the term order based on our experiences that we gained so far.