Contents - Student subdomain for University of Bath

3.3. NONLINEAR MULTIVARIATE EQUATIONS: DISTRIBUTED 93
Proposition 37 If G is a Gröbner base for I, then A l is generated, as a k-
vector space, by M := {m monomial ∈ k[x 1 , . . . , x n ]|tdeg(m) ≤ l ∧ m ∗ → G m},
i.e. the set of irreducible monomials of total degree at most l.
Note that, while M itself will depend on G (and therefore on the ordering used),
Definition 52 defines an intrinsic property of A, and hence |M| is independent
of the order chosen.
Theorem 20 ([BW93, Lemma 9.21])
( l + d
d
)
≤ H I (l) ≤
( l + n
m
Theorem 21 ([BW93, Part of Theorem 9.27]) Let G be a Gröbner base
for I ≤ k[x 1 , . . . , x n ] under a total degree order. Let I have dimension d, and
let
N = max { deg xi
(lm(g)) | g ∈ G; 1 ≤ i ≤ n } .
Then there is a unique polynomial h I ∈ Q[X] such that H I (l) = h I (l) for all
l ≥ nN. h I is called the Hilbert polynomial of I.
3.3.11 Coefficients other than fields
Most of the theory of this section, notably theorem 13, goes over to the case
when R is a P.I.D. (definition 12) rather than a field, as described in [BW93,
section 10.1]. However, when it comes to computation, things are not quite so
obvious. What is the Gröbner base of {2x, 3y} [Pau07]? There are two possible
answers to the question.
• {2x, 3y} [Tri78].
• {2x, 3y, xy} (where xy is computed as x(3y) − y(2x)) [Buc84].
We note that xy = x(3y) − y(2x) ∈ (2x, 3y), so we need xy to reduce to zero.
We therefore modify definition 42 as follows
Definition 53 Let f, g ∈ R[x 1 , . . . , x n ]. Suppose the leading coefficients of f
and g are a f and a g , and the leading monomials m f and m g . Let a be a least
common multiple of a f and a g , and write a = a f b f = a g b g . Let m be the least
common multiple of m f and m g . The S-polynomial of f and g, written S(f, g)
is defined as
m m
S(f, g) = b f f − b g g. (3.35)
m f m g
Let c f a f + c g a g = gcd(a f , a g ), and define the G-polynomial of f and g, written
G(f, g), as
m m
G(f, g) = c f f + c g g. (3.36)
m f m g
Note that (3.35) is the same as (3.24) up to a factor of gcd(a f , a g ). The S-
polynomial is defined up to unit factors, whereas the G-polynomial is much less
well-defined, but it turns out not to matter.
)
.