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3.3. NONLINEAR MULTIVARIATE EQUATIONS: DISTRIBUTED 75
Over a field, a particularly useful Gröbner base is a completely reduced
Gröbner base (abbreviated crGb) G, i.e. one where every element is completely
reduced with respect to all the others: in symbols
∀g ∈ G g ∗ ⇒ G\{g} g.
For a consistent set of linear polynomials, the crGb would be a set of linear
polynomials in one variable each, e.g. {x − 1, y − 2, z − 3}, effectively the
solution. In general, a crGb is a canonical (definition 4) form for an ideal: two
ideals are equal if, and only if, they have the same crGb (with respect to the
same ordering, of course).
Theorem 14 (Buchberger) Every polynomial ideal has a Gröbner base: we
will show this constructively for finitely-generated 8 ideals over noetherian (definition
8) rings.
Algorithm 8 (Buchberger)
Input: finite G 0 ⊂ R[x 1 , . . . , x n ]; monomial ordering >.
Output: G a Gröbner base for (G 0 ) with respect to >.
G := G 0 ; n := |G|;
# we consider G as {g 1 , . . . , g n }
P := {(i, j) : 1 ≤ i < j ≤ n}
while P ≠ ∅ do
Pick (i, j) ∈ P ;
P := P \ {(i, j)};
Let S(g i , g j ) ∗ → G h
If h ≠ 0 then
# lm(h) /∈ (lm(G))
g n+1 := h; G := G ∪ {h};
P := P ∪ {(i, n + 1) : 1 ≤ i ≤ n};
n := n + 1;
Proof. The polynomials added to G are reductions of S-polynomials of members
of G, and hence are in the same ideal as G, and therefore of G 0 . If this
process terminates, then the result satisfies condition 1, and so is a Gröbner
base for some ideal, and therefore the ideal of G. By proposition 25 and the
∗
properties of → G , h ∈ (G), so (G) is constant throughout this process and G
has to be a Gröbner base for (G 0 ). Is it possible for the process of adding new
h to G, which implies increasing (lm(G)), to go on for ever? No: corollary 1
says that the increasing chain of (lm(G)) is finite, so at some point we cannot
increase (lm(G)) any further, i.e. we cannot add a new h.
8 In fact, every polynomial ideal over a noetherian ring is finitely generated. However,
it is possible to encode undecidability results in infinite descriptions of ideals, hence we say
“finitely generated” to avoid this paradox.