| ## 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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