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Signature-Based Gr\"obner Basis Algorithms --- Extended MMM Algorithm for computing Gr\"obner bases
Signature-based algorithms is a popular kind of algorithms for computing
Gr\"obner bases, and many related papers have been published recently. In this
paper, no new signature-based algorithms and no new proofs are presented.
Instead, a view of signature-based algorithms is given, that is,
signature-based algorithms can be regarded as an extended version of the famous
MMM algorithm. By this view, this paper aims to give an easier way to
understand signature-based Gr\"obner basis algorithms
The F5 Algorithm in Buchberger's Style
The famous F5 algorithm for computing \gr basis was presented by Faug\`ere in
2002. The original version of F5 is given in programming codes, so it is a bit
difficult to understand. In this paper, the F5 algorithm is simplified as F5B
in a Buchberger's style such that it is easy to understand and implement. In
order to describe F5B, we introduce F5-reduction, which keeps the signature of
labeled polynomials unchanged after reduction. The equivalence between F5 and
F5B is also shown. At last, some versions of the F5 algorithm are illustrated
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