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Buchberger's algorithm
About: Buchberger's algorithm is a research topic. Over the lifetime, 129 publications have been published within this topic receiving 6330 citations.
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TL;DR: In this paper, the Buchberger algorithm is replaced by the simultaneous reduction of several polynomials, which is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods.
1,089 citations
10 Jul 2002
TL;DR: In this article, the Buchberger criteria were replaced by an optimal criteria and the resulting algorithm (called F5) was shown to generate no useless critical pairs if the input is a regular sequence.
Abstract: This paper introduces a new efficient algorithm for computing Grobner bases. We replace the Buchberger criteria by an optimal criteria. We give a proof that the resulting algorithm (called F5) generates no useless critical pairs if the input is a regular sequence. This a new result by itself but a first implementation of the algorithm F5 shows that it is also very efficient in practice: for instance previously untractable problems can be solved (cyclic 10). In practice for most examples there is no reduction to zero. We illustrate this algorithm by one detailed example.
774 citations
TL;DR: The lexicographical GroBner basis can be obtained by applying this algorithm after a total degree Grobner basis computation: it is usually much faster to compute the basis this way than with a direct application of Buchberger's algorithm.
708 citations
TL;DR: This is the English translation (by Michael P. Abramson) of the PhD thesis of Bruno Buchberger, in which he introduced the algorithmic theory of Grobner bases.
673 citations
28 Mar 1983
TL;DR: In this article, the authors give upper and lower bounds for the degrees of the elements of a Gr6bner base, which are based on projective algebraic geometry and the choice of the ordering appears to be critical.
Abstract: In the past few years, two very different methods have been developed for solving systems of algebraic equations : the method of Gr6bner bases or standard bases [Buc I, Buc 2, Tri, P.Y] and the one which I presented in Eurosam 79 [Laz 2, Laz 3] based on gaussian elimination in some matrices. Although they look very different, they are, in fact, very similar, at least if we restrict ourselves to the first step of my method. On the other hand Gr6bner base algorithms are very close to the tangent cone algorithm of Mora [Mor]. All of these algorithms are related to Gaussian elimination. In the first part of this paper, we try to develop all these relations and to show that this leads to improvements in some of these algorithms. In the second part we give upper and lower bounds for the degrees of the elements of a Gr6bner base. These bounds are based on projective algebraic geometry. The choice of the ordering appears to be critical : lexicographical orderings give Gr6bner bases of high degree, while reverse lexicographical orderings lead to low degrees.
431 citations