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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 {L(f) | f ∈ I}. Then f1,,fs I is called a standard basis of I if L(f1),,L(fs) generate L(I).

Properties

normal form:
A function NF : Rr ×{GG  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:
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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