Gröbner bases computation using syzygies

2007

In the computation of Groebner bases using Buchberger's Algorithm, a key issue for improving the efficiency is to create techniques to help us avoid as many unnecessary pairs of polynomials from the non-computed set of pairs as possible. A good solution would be to avoid those pairs that can be easily ignored without computing their S-polynomials, and hence to process only on the set of pairs of generators of the module generated by syzygies. This paper details an improvment of Buchberger's Algorithm for computing Groebner bases by defining the module of solutions of a homogeneous linear equation with polynomial coefficients (called the syzygy module). As a consequence, we use these syzygy modules to give another equivalent condition for a set to be a Groebner basis for an ideal. As a result we demonastrate that this new condition can significantly improve the Buchberger's Algorithm to compute Groebner bases.

Electronic Proceedings in Theoretical Computer Science, 2019

In this paper, we make a contribution to the computation of Gröbner bases. For polynomial reduction, instead of choosing the leading monomial of a polynomial as the monomial with respect to which the reduction process is carried out, we investigate what happens if we make that choice arbitrarily. It turns out not only this is possible (the fact that this produces a normal form being already known in the literature), but, for a fixed choice of reductors, the obtained normal form is the same no matter the order in which we reduce the monomials. To prove this, we introduce reduction machines, which work by reducing each monomial independently and then collecting the result. We show that such a machine can simulate any such reduction. We then discuss different implementations of these machines. Some of these implementations address inherent inefficiencies in reduction machines (repeating the same computations). We describe a first implementation and look at some experimental results.

Proceedings of the 1993 international symposium on Symbolic and algebraic computation - ISSAC '93, 1993

2016

Let S = k[x1, x2, . . . , xn] denote a polynomial ring over a field k where x1, x2, . . . , xn are indeterminates. A Gröbner basis is a set of polynomials in S which has several remarkable properties which enable us to carry out standard operations on ideals, rings and modules in an algorithmic way. Every set of polynomials in S can be transformed into a Gröbner basis. This process generalises three important algorithms: (1) Gauss elimination method for solving a system of linear equations, (2) Euclid’s algorithm for finding the greatest common divisor and (3) The simplex method of linear programming. One of the goals of these two lectures is to explain how to reduce the problem of solving a system of polynomial equations to a problem of finding eigenvalues of commuting matrices. We will introduce term orders first on the set of monomials in S and define the concept of Gróbner basis of an ideal. Term orders on monomials in k[x1, x2, . . . , xn] The set of monomials in the polynomial...

In Grobner bases computation a general open question is how to guide calculations coping with numerical coecients and/or not exact input data. It may happen that, due to error accumulation or insucient working precision, the result is not one theoretically expects. The basis may have more or less polynomials, a dierent number of solutions, a zero set with wrong multiplicity, and so on. Augmenting precision we may overcome algorithmic errors, but we don't know in advance how much it should be, and a trial-and-error approach is often the only way. Coping with initial errors is an even more dicult task. In this work the combined use of syzygies and interval arithmetic is proposed as a technique to decide at each critical point of the algorithm what to do.

1993

Almost every Computer Algebra System contains some implementation of the Gr obner bases algorithm. The present implementation has the following speci c features:

HAL (Le Centre pour la Communication Scientifique Directe), 2022

International Virtual FDP on “Frontiers of Mathematics”, 2020

* Introduction ? Varieties ? Ideals ? Linear Case ? Polynomials of One Variable * Gröbner Bases ? Term Orders ? S-Polynomials ? Buchberger’s Algorithm ? Sample Computations * Some Application of Gröbner Bases ? The 3-Color Problem ? Automatic Geometric Theorem Proving ? Other Applications * References

Lecture Notes in Computer Science, 2006

In Gröbner bases computation, as in other algorithms in commutative algebra, a general open question is how to guide the calculations coping with numerical coefficients and/or not exact input data. It often happens that, due to error accumulation and/or insufficient working precision, the obtained result is not one expects from a theoretical derivation. The resulting basis may have more or less polynomials, a different number of solution, roots with different multiplicity, another Hilbert function, and so on. Augmenting precision we may overcome algorithmic errors, but one does not know in advance how much this precision should be, and a trial-and-error approach is often the only way to follow. Coping with initial errors is an even more difficult task. In this experimental work we propose the combined use of syzygies and interval arithmetic to decide what to do at each critical point of the algorithm.

Arxiv preprint arXiv:1101.3589, 2011

This paper describes a Buchberger-style algorithm to compute a Gröbner basis of a polynomial ideal, allowing for a selection strategy based on "signatures". We explain how three recent algorithms can be viewed as different strategies for the new algorithm, and how other selection strategies can be formulated. We describe a fourth as an example. We analyze the strategies both theoretically and empirically, leading to some surprising results.