It is known that the sequence, in which pairs are considered for S-polynomial computation, has an impact on the computation time needed for computing the Gröbner basis. This is particularly true when lexical ordering for terms is used.
Usually, pairs are ordered w.r.t. the least common multiples (lcm) of the leading terms of the polynomials in the pair. Pairs, whose lcm is smaller in a particular term ordering are considered first. A canonic choice for the term ordering used for pairs is to use the same ordering as for terms in the polynomials, but this need not in all examples be the best choice. In fact, it turned out that - for lexical ordering of terms - total degree ordering for pairs is in some examples significantly faster than lexical ordering whereas in other examples the opposite is true. See also [4] and [5].
Table 1 shows timings for the input as given in Examples 6 and 8.
Table 1: Timings in seconds on an HP9000/725 workstation.