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C.1 Standard bases
Definition
Let
623#623 and let 253#253 be a submodule of 624#624.
Note that for r=1 this means that 253#253 is an ideal in 53#53.
Denote by 625#625 the submodule of 624#624 generated by the leading terms
of elements of 253#253, i.e. by
626#626.
Then
627#627 is called a standard basis of 253#253
if
628#628 generate 625#625.
A standard basis is minimal if
629#629.
A minimal standard basis is completely reduced if
630#630
Properties
- normal form:
-
A function
631#631, is called a normal
form if for any 632#632 and any standard basis 190#190 the following
holds: if
633#633 then 148#148 does not divide
634#634 for all 256#256.
The function may also be applied to any generating set of an ideal:
the result is then not uniquely defined.
635#635 is called a normal form of 23#23 with
respect to 190#190 - ideal membership:
-
For a standard basis 190#190 of 253#253 the following holds:
276#276 if and only if
636#636.
- Hilbert function:
- Let
637#637 be a homogeneous module, then the Hilbert function
638#638 of 253#253 (see below)
and the Hilbert function 639#639 of the leading module 625#625
coincide, i.e.,
640#640.
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