## A New Efficient Algorithm for Computing Gröbner Bases (F4) (2002)

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Venue: | IN: ISSAC ’02: PROCEEDINGS OF THE 2002 INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND ALGEBRAIC COMPUTATION |

Citations: | 269 - 54 self |

### BibTeX

@INPROCEEDINGS{Faugère02anew,

author = {Jean-charles Faugère},

title = {A New Efficient Algorithm for Computing Gröbner Bases (F4)},

booktitle = {IN: ISSAC ’02: PROCEEDINGS OF THE 2002 INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND ALGEBRAIC COMPUTATION},

year = {2002},

pages = {75--83},

publisher = {}

}

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### Abstract

This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo computations.

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Citation Context ...� to recover the value © in . In fact the iteration may be stopped when ¦ �� � becomes stable (see [S.R71] for multi modular methods which may be generalized to¥ § adic method). The global complexity =-=[Knu81]-=- of this algorithms is �� £��¤ �� ¨s� Bareiss methods��¤�� � ��¤ �� ¨ ����¤ �� ¨ �� multi modular method �£ ��¤ �� ¨ �£ � ¥ -adic method ��¤ Note that -adic method is also an iterative algorithm (in f... |

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Citation Context ... more powerful. Historically, the Buchberger algorithm was the first algorithm for computing such Gröbner bases. It may eventually be possible to suggest two improvements for the Buchberger algorithm =-=[3, 4, 5]-=-. The first improvement is concerned with strategies: during a Gröbner computation, several choices can be made (select a critical pair, choose a reductor) this aspect of the problem is not directly s... |

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Citation Context ... syzygy if v(g) = 0. The so called principal syzygies si,j = fjFi−fiFj are syzygies. The set of all syzygies is a module and abbreviated by Syz (for more information on syzygies we refer to [2] or to =-=[1]-=-). Let PSyz be the module generated by the principal syzygies. For a generic (random) polynomial system (f1, . . . , fm), Syz = PSyz. We can extend the admissible ordering < to P m with the following ... |

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Citation Context ...rithm, one has a lot of choices: select a critical pair in the list of critical pairs. ¦ choose one reductor among a list of reductors when reducing a polynomial by a list of ¦ polynomials. Buchberger=-=[Buc65]-=- proves that these choices are not important for the correctness of the algorithm, but it is well known that these choices are crucial for the total time computation. Moreover the best strategies [Gio... |

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Citation Context ...CO LTR 21.024. 1seventually be possible to suggest two improvements: sinces¡ ¢ of the times is spent computing zero it would be useful to have more powerful criterion to remove useless critical pairs =-=[Buc79]-=- (a powerful theoretical criteria exists but it is too costly); this crucial aspect of the problem is not studied in this paper, but is implemented in another algorithm (£ £ [Fau98a]). The second impr... |

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Citation Context ...e Buchberger Criteria. Since it is not the subject of this paper to improve the Buchberger Criteria we will use a standard implementation of these criteria such as the Gebauer and Moller installation =-=[GM88]-=-: Buchberger Criteria Specification: Input: � ¢ �§¨ � � �§¨ � �¨ £ ¥ £ � � ¡ � ¢ © �� � � © �� �¡ � ¦ ��§ ¨ © �¥� �£���¥ � ¢ © � � ¤ ¥¦ § �� � � �§¥� �� ���¥��§¤ �§ �� ¤��¥ §© � �� �§¤ � § ��� ��� � ¥... |

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Citation Context .... The second improvement is concerned with strategies: during a Gröbner computation, several choices can be made (select a critical pair, choose a reductor) and even if strategies have been proposed (=-=[Gio91]-=- or even [Ger95]) the heuristics which they rely on could not be satisfactorily explained. So it is difficult to be convinced that they are optimal optimizations. Another bad consequence is that it is... |

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Citation Context ... total degree (Definition 2.6) are� £ ), then we call ¢ � a (truncated) £ -Gröbner basis of ¢ . The following theorem give a structure to this list when the polynomials are homogenenous. Theorem 2.5 (=-=[Laz83]-=-, [Bec93] p. 471) For homogeneous polynomials� © � � � � � � � , ¢ � is a Gröbner basis “up to degree £ ” that is to say: � §¨ ¦ ¥sis well defined for polynomials � such that � � � �� �s£ . ¦ ¡ ¥ � ¢ ... |

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Citation Context ...global complexity for solving the system iss�� �� �� instead ofs��¤� . When the matrix has a regular pattern it is possible to apply even more efficient techniques (for Toeplitz matrices for instance =-=[Bin94]-=-). These methods can not be applied in our case since the pattern of the generated matrices is not regular enough. A significant drawback of these methods is that there is no speedup if we try to solv... |

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Citation Context ...resenting the new algorithm. This section has been divided into several parts: first (2.1), we review the necessary mathematical notations (we make the choice to use the same notations as in the book =-=[Bec93]-=-) and in 2.2 we establish the link between linear algebra (matrices) and polynomial algebra. Then we present (2.3) a basic version of the algorithm without any criteria to eliminate useless pairs. A i... |

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Citation Context ...ess critical pairs [Buc79] (a powerful theoretical criteria exists but it is too costly); this crucial aspect of the problem is not studied in this paper, but is implemented in another algorithm (£ £ =-=[Fau98a]-=-). The second improvement is concerned with strategies: during a Gröbner computation, several choices can be made (select a critical pair, choose a reductor) and even if strategies have been proposed ... |

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Citation Context ... other words, (g1, . . . , gm) is a syzygy. Several papers investigate those issues: Buchberger [4] proposes two criteria to remove a lot of useless critical pairs; staggered linear bases are used in =-=[7]-=-; the idea of [10] is to compute simultaneously a Gröbner basis and a basis of the module of syzygies: a critical pair is not considered if the corresponding syzygy is a linear combination of some ele... |

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Citation Context ...the input matrix� must be non singular or square or positive definite. On the other hand, parallel algorithms and parallel implementations exist [Van93, Hea, Don, Alv, Pey]. We have modified the smms =-=[Alv93]-=- program in order to work with modulo¥ coefficients in order to evaluate the costs of different algorithms. Solving large linear equations modulo a small prime number has been studied in cryptology [L... |

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Citation Context ...f methodologies in the field of benchmark for linear system solving. We adopt the following points: 19s1. We compare the new algorithm with state of the art Gröbner bases implementation (namely Magma =-=[Can98]-=-, PoSSo/Frisco Gröbner engine [Tra97], Macaulay 2 [Sti89, Gra97], Singular [Gre97], Asir [Tak96], Cocoa, Axiom, Maple [Cha91] and Gb [Faub]). It is also crucial to compare the implementation of the ne... |

3 |
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Citation Context ...can be solved, the algorithms are competitive with numerical methods. The main conclusion to be drawn from practice and experience of solving polynomial systems coming from various fields (industrial =-=[Fau98b]-=- problems, pure mathematics [Fro96]) is the following: first of all, even though the computation of a Gröbner basis is a crucial point it must be emphasized that it is only one step in the full solvin... |

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Benchmarks for polynomial solvers. avalaible on the WEB http://posso.lip6.fr/˜jcf/benchs.html
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Citation Context ... high level language/compiler. £ ¤ has been implemented in � and most of the competitors are implemented in C/C++. 2. The list of examples is also a crucial issue: the examples can be easily accessed =-=[Faua]-=- (the web site contains pointers to the Frisco test suite). The list is composed of classical benchmarks (Cyclic � , Katsura � ) but also of industrial examples (coming from signal theory, robotics). ... |

1 |
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Citation Context ...state of the art Gröbner bases implementation (namely Magma [Can98], PoSSo/Frisco Gröbner engine [Tra97], Macaulay 2 [Sti89, Gra97], Singular [Gre97], Asir [Tak96], Cocoa, Axiom, Maple [Cha91] and Gb =-=[Faub]-=-). It is also crucial to compare the implementation of the new algorithm with the Buchberger algorithm implemented by the same person (in this case the author). In our opinion it is also important to ... |

1 |
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Citation Context ...mpetitive with numerical methods. The main conclusion to be drawn from practice and experience of solving polynomial systems coming from various fields (industrial [Fau98b] problems, pure mathematics =-=[Fro96]-=-) is the following: first of all, even though the computation of a Gröbner basis is a crucial point it must be emphasized that it is only one step in the full solving process (change of ordering, tria... |

1 |
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Citation Context ...rovement is concerned with strategies: during a Gröbner computation, several choices can be made (select a critical pair, choose a reductor) and even if strategies have been proposed ([Gio91] or even =-=[Ger95]-=-) the heuristics which they rely on could not be satisfactorily explained. So it is difficult to be convinced that they are optimal optimizations. Another bad consequence is that it is very difficult ... |

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