2. Sequential Algorithms

On the algorithmic side, our new approach differs from the former in the following point.

For any Groebner or standard basis computation the efficiency of an algorithm depends essentially on the strategy how to chose the next pair from the pairset constructed so far. The major criteria to sort (and select from) a pairset are usually order-dependend ( O) (the order of the leading term of the s-polynomial or of the syzygy of the initial monomials) or the module index ( M) for implementations of free resolutions. Moreover, there are some secondary criteria like the length of polynomials etc.

Of course, the organization of the pairset depends also on the sequence ( S) with which new generators are added to the set of known generators. The following table gives an overview about the use of the major types of known criteria within the different algorithms:

	Naive	Schreyer	LaScala	Hilbert-driven	Sequential
1. criterion	M	M	O	O	S
2. criterion	O	O	M	M
3. criterion	S	S	S	S

For a description of these algorithms see [7] (for more details also [1] and [5]).

Definition 2.1   An algorithm for the computation of a free resolutions is called sequential if it uses the sequence of generators as first criterion for ordering the pairsets. This means, it extends recursively a resolution for the first $n-1$ generators to a resolution of $n$ generators of a given ideal.