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A Generalization of Gröbner Basis Algorithms to Polycyclic Group Rings

✍ Scribed by Klaus Madlener; Birgit Reinert

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
656 KB
Volume
25
Category
Article
ISSN
0747-7171

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✦ Synopsis

It is well-known that for the integral group ring of a polycyclic group several decision problems are decidable, in particular the ideal membership problem. In this paper we define an effective reduction relation for group rings over polycyclic groups. This reduction is based on left multiplication and hence corresponds to left ideals. Using this reduction we present a generalization of Buchberger's Gröbner basis method by giving an appropriate definition of "Gröbner bases" in this setting and by characterizing them using the concepts of saturation and s-polynomials. The approach is extended to two-sided ideals and a discussion on a Gröbner bases approach for right ideals is included.

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