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C.1 Standard bases
Definition
Let
621#621 and let 251#251 be a submodule of 622#622.
Note that for r=1 this means that 251#251 is an ideal in 53#53.
Denote by 623#623 the submodule of 622#622 generated by the leading terms
of elements of 251#251, i.e. by
624#624.
Then
625#625 is called a standard basis of 251#251
if
626#626 generate 623#623.
A standard basis is minimal if
627#627.
A minimal standard basis is completely reduced if
628#628
Properties
- normal form:
-
A function
629#629, is called a normal
form if for any 630#630 and any standard basis 189#189 the following
holds: if
631#631 then 148#148 does not divide
632#632 for all 254#254.
The function may also be applied to any generating set of an ideal:
the result is then not uniquely defined.
633#633 is called a normal form of 23#23 with
respect to 189#189 - ideal membership:
-
For a standard basis 189#189 of 251#251 the following holds:
274#274 if and only if
634#634.
- Hilbert function:
- Let
635#635 be a homogeneous module, then the Hilbert function
636#636 of 251#251 (see below)
and the Hilbert function 637#637 of the leading module 623#623
coincide, i.e.,
638#638.
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