104 research outputs found
A survey on signature-based Gr\"obner basis computations
This paper is a survey on the area of signature-based Gr\"obner basis
algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain
the general ideas behind the usage of signatures. We show how to classify the
various known variants by 3 different orderings. For this we give translations
between different notations and show that besides notations many approaches are
just the same. Moreover, we give a general description of how the idea of
signatures is quite natural when performing the reduction process using linear
algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table
Predicting zero reductions in Gr\"obner basis computations
Since Buchberger's initial algorithm for computing Gr\"obner bases in 1965
many attempts have been taken to detect zero reductions in advance.
Buchberger's Product and Chain criteria may be known the most, especially in
the installaton of Gebauer and M\"oller. A relatively new approach are
signature-based criteria which were first used in Faug\`ere's F5 algorithm in
2002. For regular input sequences these criteria are known to compute no zero
reduction at all. In this paper we give a detailed discussion on zero
reductions and the corresponding syzygies. We explain how the different methods
to predict them compare to each other and show advantages and drawbacks in
theory and practice. With this a new insight into algebraic structures
underlying Gr\"obner bases and their computations might be achieved.Comment: 25 pages, 3 figure
The F5 Criterion revised
The purpose of this work is to generalize part of the theory behind Faugere's
"F5" algorithm. This is one of the fastest known algorithms to compute a
Groebner basis of a polynomial ideal I generated by polynomials
f_{1},...,f_{m}. A major reason for this is what Faugere called the algorithm's
"new" criterion, and we call "the F5 criterion"; it provides a sufficient
condition for a set of polynomials G to be a Groebner basis. However, the F5
algorithm is difficult to grasp, and there are unresolved questions regarding
its termination.
This paper introduces some new concepts that place the criterion in a more
general setting: S-Groebner bases and primitive S-irreducible polynomials. We
use these to propose a new, simple algorithm based on a revised F5 criterion.
The new concepts also enable us to remove various restrictions, such as proving
termination without the requirement that f_{1},...,f_{m} be a regular sequence.Comment: Originally submitted by Arri in 2009, with material added by Perry
since 2010. The 2016 editions correct typographical issues not caught in
previous editions bring the theory of the body into conformity with the
published version of the pape
A survey on signature-based algorithms for computing Gröbner basis computations
International audienceThis paper is a survey on the area of signature-based Gröbner basis algorithms that was initiated by Faugère's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research
The F5 Criterion Revised
The purpose of this work is to generalize part of the theory behind Faugere\u27s F5 algorithm. This is one of the fastest known algorithms to compute a Gröbner basis of a polynomial ideal I generated by polynomials f1,…,fm. A major reason for this is what Faugere called the algorithm\u27s new criterion, and we call the F5 criterion : it provides a sufficient condition for a set of polynomialsGto be a Gröbner basis. However. the F5 algorithm is difficult to grasp, and there are unresolved questions regarding its termination. This paper introduces some new concepts that place the criterion in a more general setting:S-Gröbner bases and primitive S-irreducible polynomials. We use these to propose a new, simple algorithm based on a revised F5 criterion. The new concepts also enable us to remove various restrictions, such as proving termination without the requirement that f1,…,fm be a regular sequence. (C) 2011 Elsevier Ltd. All rights reserved
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