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3.3. NONLINEAR MULTIVARIATE EQUATIONS: DISTRIBUTED 81
Proposition 33 (Buchberger’s gcd (or First) Criterion [Buc79]) If
then S(f, g) ∗ → {f,g} 0.
gcd(lm(f), lm(g)) = 1, (3.25)
In practice, this is implemented by not even adding to P , either in the initial
construction or when augmenting it due to a new h, pairs for which equation
(3.25) is satisfied. This proposition also explains why the more general construct
of an S-polynomial is not relevant to linear equations: when f and g are linear,
if they have the same leading variable, one can reduce the other, and if they do
not, then the S-polynomial reduces to zero.
Proposition 34 (Buchberger’s lcm (or Third) Criterion [Buc79]) If I =
(B) contains f, g, h and the reductions under ∗ → B of S(f, g) and S(f, h), and
if both lcm(lm(f), lm(g)) and lcm(lm(f), lm(h)) divide lcm(lm(g), lm(h)), then
S(g, h) ∗ → B 0, and hence need not be computed.
This has been generalised to a chain of polynomials f i connecting g and h: see
[BF91].
Propositions 33 and 34 are therefore sufficient to say that we need not compute
an S-polynomial: the question of whether they are necessary is discussed
by [HP07]. Terminology varies in this area, and some refer to Buchberger’s
Second Criterion as well. The more descriptive gcd/lcm terminology is taken
from [Per09].
Whereas applying the gcd Criterion (Proposition 33) to S(f, g) depends only
on f and g, applying the lcm Criterion (Proposition 34 and its generalisations)
to S(g, h) depends on the whole past history of the computation. It might be
that the Criterion is not applicable now, but might become applicable in the
future. Hence we can ask
which way of picking elements from P in Algorithm 8 will maximise
the effectiveness of the lcm Criterion?
A partial answer was given in [Buc79].
Definition 46 We say that an implementation of Algorithm 8 follows a normal
selection strategy if, at each stage, we pick a pair (i, j) such that lcm(lm(g i ), lm(g j ))
is minimal with respect to the ordering in use.
This does not quite specify the selection completely: given a tie between (i, j)
and (i ′ , j ′ ) (with i < j, i ′ < j ′ ), we choose the pair (i, j) if j < j ′ , otherwise
(i ′ , j ′ ) [GMN + 91].
For the sake of simplicity, let us assume that we are dealing with a total
degree ordering, or a lexicographic ordering. The case of weighted orderings,
or so-called multi-graded orderings (e.g. an elimination ordering each of whose
components is a total degree ordering) is discussed in [BCR11].