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The remark that Buchberger's algorithm is (not considering any refinements thereof) the only known way to compute Groebner bases is not correct.
Another approach that has been implemented (and that has been found to be very competitive in terms of running time) is based on the concept of involutive bases. The latter are based on ideas from differential algebra, in particular on work from the french mathematician Riquier. Involutive bases have been investigated, among others, by Gerdt and Blinkov.
62.214.243.240 20:11, 17 June 2006 (UTC)
This article should include a discussion with examples about the complete reduction of S(i,j) by G and I wish to add this.Youriens (talk) 18:28, 8 April 2021 (UTC)
- As the article is presently written, this is the complete reduction that is used. The lacking content is that one gets an correct algorithm by using lead reduction instead of complete reduction, and doing complete reduction only after the termination of the main loop. Be free to clarify the equivalence of these two variants. D.Lazard (talk) 19:51, 8 April 2021 (UTC)
Do you think it would be useful to others to have a complete example of using Buchberger's algorithm to compute a reduced basis either in this article or the Grobner Basis article? I think I can write one but likely would need some further review and corrections by others more familiar with the subject. I've found a number of descriptions and examples on the net but none in my opinion which do a nice job of fully explaining the process with a completely worked-out example. This is the best one I found on Physics Forum: [1]