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3.3. NONLINEAR MULTIVARIATE EQUATIONS: DISTRIBUTED 73
Definition 40 If lm(g) divides lm(f), then we say that g reduces f to h =
lc(g)f − (lt(f)/lm(g))g, written f → g h. Otherwise we say that f is reduced
with respect to g. The Maple user should note that Maple’s Reduce command
actually implements complete reduction — see Definition 41.
If R is a field, division is possible, and so it is more usual to reduce f to
f − (lt(f)/lt(g))g. In the construction of h, the leading terms of both lc(g)f
and (lt(f)/lm(g))g are lc(f)lc(g)lm(f), and so cancel. Hence lm(h) < lm(f).
This observation and theorem 3 give us the following result.
Proposition 23 Any chain f 1 → g f 2 → g f 3 · · · is finite, i.e. terminates in a
polynomial h reduced with respect to g. We write f 1
∗
→
g
h.
These concepts and results extend to reduction by a set G of polynomials, where
f → G h means ∃g ∈ G : f → g h. We must note that a polynomial can have
several reductions with respect to G (one for each element of G whose leading
monomial divides the leading monomial of f). For example, let G = {g 1 =
x − 1, g 2 = y − 2} and f = xy. Then there are two possible reductions of f:
f → g1 h 1 = f − yg 1 = y, and f → g2 h 2 = f − xg 2 = 2x. In this case h 1 → g2 2
and h 2 → g1 2, so that f→ ∗ G 2 uniquely, but even this need not always be the case.
If we let G = {g 1 = x−1, g 2 = x 2 } and f = x 2 −1, then f → g2 h 2 = f −g 2 = −1,
whereas f → g1 f − xg 1 = x − 1 → g1 0: so f ∗ → G 0 or −1.
This definition deals with reduction of the leading monomial of f by g, but
it might be that other monomials are reducible. For simplicity we consider the
case when R is a field.
Definition 41 If any term cm of f is reducible by g, i.e. the leading monomial
of g divides m, we say that g part-reduces f, and write f ⇒ g f − (cm/lt(g))g.
We can continue this process (only finitely often, by repeated application of
theorem 3), until no monomial of f is reducible by g, when we write f ∗ ⇒ g h, and
say that f is completely reduced by g to h. Again, this extends to reduction by
a set of polynomials.
In section 3.2.1, we performed row operations: subtracting a multiple of one
row from another, which is essentially what reduction does, except that the
‘multiple’ can include a monomial factor. It turns out that we require a more
general concept, given in the next definition.
Definition 42 Let f, g ∈ R[x 1 , . . . , x n ]. The S-polynomial of f and g, written
S(f, g) is defined as
S(f, g) =
lt(g)
gcd(lm(f), lm(g)) f − lt(f)
g. (3.24)
gcd(lm(f), lm(g))
We note that the divisions concerned are exact, and that this generalises reduction
in the sense that, if lm(g) divides lm(f), then f → g S(f, g). As with
reduction, the leading monomials in the two components on the righthand
side of equation (3.24) cancel. Another way of thinking of the S-polynomial