|  |  C.1 Standard bases 
  Definition  Let![$R = \hbox{Loc}_< K[\underline{x}]$](sing_584.png) 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 
![$I \subseteq K[\underline{x}]^r$](sing_598.png) 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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