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A completion procedure for computing a canonical basis for a k- subalgebra. (English) Zbl 0692.13001
Computers and mathematics, Proc. Conf., Cambridge/Mass. 1989, 1-11 (1989).
[For the entire collection see Zbl 0671.00018.]
A procedure and related theory for computing a canonical basis for a finitely presented k-subalgebra are presented. With a slight modification, the procedure can also be used for the membership problem of a unitary subring generated by a finite basis using a Gröbner basis like approach. In contrast to Shannon and Sweedler’s approach using tag variables, this approach is direct.
In section 2, definitions are given. Section 3 discusses how rules are made from polynomials, and a reduction relation is defined using a set of rules corresponding to a k-subalgebra basis. A canonical basis of a k- subalgebra is defined. Section 4 defines superpositions, critical pairs and S-polynomials which lead to a finite test for checking whether a given finite basis of a k-subalgebra is a canonical basis. Section 5 is the main result which shows that if all S-polynomials corresponding to critical pairs of a set of rules reduce to 0, then the corresponding basis is canonical. Section 6 outlines a completion procedure based on the test of section 5, and properties of canonical bases generated by a completion procedure are discussed. The procedure is illustrated using examples. Some comments on how this approach can be modified to be applicable to unitary subrings are given in the final section.
Reviewer: Y.Kuo

MSC:
13-04 Software, source code, etc. for problems pertaining to commutative algebra
13B99 Commutative ring extensions and related topics
68W30 Symbolic computation and algebraic computation