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C.1 Standard bases
Definition
Let
and let be a submodule of .
Note that for r=1 this means that is an ideal in .
Denote by the submodule of generated by the leading terms
of elements of , i.e. by
.
Then
is called a standard basis of
if
generate .
A standard basis is minimal if
.
A minimal standard basis is completely reduced if
Properties
- normal form:
-
A function
, is called a normal
form if for any and any standard basis the following
holds: if
then does not divide
for all .
The function may also be applied to any generating set of an ideal:
the result is then not uniquely defined.
is called a normal form of with
respect to - ideal membership:
-
For a standard basis of the following holds:
if and only if
.
- Hilbert function:
- Let
be a homogeneous module, then the Hilbert function
of (see below)
and the Hilbert function of the leading module
coincide, i.e.,
.
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