|
B.2.4 Local orderings
For ls, ds, Ds and, if the weights are positive integers, also for ws and
Ws, we have
Loc K[x] = K[x](x),
the localization of
K[x]
at the maximal ideal
(x1,...,xn).
- ls:
negative lexicographical ordering:
xα < xβ ⇔∃ 1 ≤ i ≤ n : α1 = β1,…,αi−1 = βi−1,αi > βi.
- ds:
negative degree reverse lexicographical ordering:
let
deg(xα) = α1 + + αn, then
xα < xβ ⇔ deg(xα) > deg(xβ) or
deg(xα) = deg(xβ) and
∃ 1 ≤ i ≤ n : αn = βn,…,αi+1 = βi+1,αi > βi.
- Ds:
negative degree lexicographical ordering:
let deg(xα) = α1 + + αn, then
xα < xβ ⇔ deg(xα) > deg(xβ)
or
deg(xα) = deg(xβ) and ∃ 1 ≤ i ≤ n : α1 = β1,…,αi−1 = βi−1,αi < βi.
- ws:
(general) weighted reverse lexicographical ordering:
ws(w1,…,wn), w1
a nonzero integer,
w2,…,wn
any integer (including 0),
is defined as ds
but with
deg(xα) = w1α1 + + wnαn.
- Ws:
(general) weighted lexicographical ordering:
Ws(w1,…,wn), w1
a nonzero integer,
w2,…,wn
any integer (including 0),
is defined as Ds
but with
deg(xα) = w1α1 + + wnαn.
|