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.

Wed Sep 2 09:42:51 MDT 1998