| ## 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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