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C.1 Standard bases
Definition
Let R = Loc<K[x] and let I be a submodule of Rr. Note that for r=1 this means that I is an ideal
in R. Denote by L(I) the submodule of Rr generated by the leading terms of elements of I, i.e. by
. Then f1,…,fs ∈ I is called a standard basis of I if L(f1),…,L(fs) generate L(I).
Properties
- normal form:
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A function
NF : Rr ×{G∣G a standard basis}→ Rr,(p,G)
NF(p|G), is called a normal form if for any p ∈ Rr and any
standard basis G the following holds: if NF(p|G) ⁄= 0 then L(g) does not divide L(NF(p|G)) for all
g ∈ G.
NF(p|G) is called a normal form of p with respect to G (note that such a function is not unique).
- ideal membership:
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For a
standard basis G of I the following holds: f ∈ I if and only if NF(f,G) = 0.
- Hilbert function:
- Let I ⊆ K[x]r be a
homogeneous module, then the Hilbert function HI of I (see below) and the Hilbert function HL(I) of the
leading module L(I) coincide, i.e., HI = HL(I).
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