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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,i1 = βi1i > 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,i+1 = βi+1i > 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,i1 = βi1i < 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.


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