Groebner Bases and an Improvement on Buchberger's 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.

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...

Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92, 1992

Grobner bases are an important tool. Therefore the mainComputerAl gebraSy stems contain procedures for computing such ideal bases. These algorithms typically spend much time in reducing so called " S-polynomials" to O. One of Buchberger's two criteria for avoiding superfluous reductions to O was interpreted in paper by G ebauer & iVIoller [1988) as a criterion for finding in a, generating system of particular syzygies redundant ones. In the present paper this idea is extended. While the Grobner basis is constructed, a set of syzygies is updated and an " S-polynomial" is not considered if its coresponcling syzygy turns out to depend on the already known syzygies. Using this concept, we cover both of Buchberger's criteria, are able to include information on polynomials (i.e. syzygies) either obtained from input or from intermediate calculations, ancl avoid more superfluous reductions. We illustrate this procedure by two examples. * This reseal ch was made in the preliminary part of the research fiTI.Ln(. ecl with the CEC Basic Research ESPRIT contract n. 6846

2009

In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Groebner bases via Buchberger's Algorithm.

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

International Journal of Engineering and Advanced Technology, 2019

This paper is a survey on Groebner basis and its applications on some areas of Science and Technology. Here we have presented some of the applications of concepts and techniques from Groebner basis to broader area of science and technology such as applications in steady state detection of chemical reaction network (CRN) by determining kinematics equations in the investigation and design of robots. Groebner basis applications could be found in vast area in circuits and systems. In pure mathematics, we can encounter many problems using Groebner basis to determine that a polynomial is invertible about an ideal, to determine radical membership, zero divisors, hence so forth. A short note is being presented on Groebner basis and its applications.

2007

We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Groebner basis in Z/m[x_1,x_2,...,x_n] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.

Journal of Symbolic Computation, 1996

In this paper we will define analogs of Gröbner bases for R-subalgebras and their ideals in a polynomial ring R[x 1 ,. .. , xn] where R is a noetherian integral domain with multiplicative identity and in which we can determine ideal membership and compute syzygies. The main goal is to present and verify algorithms for constructing these Gröbner basis counterparts. As an application, we will produce a method for computing generators for the first syzygy module of a subset of an R-subalgebra of R[x 1 ,. .. , xn] where each coordinate of each syzygy must be an element of the subalgebra.

2007

We present an algorithm for computing Gröbner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Möller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Möller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Gröbner bases to the problem of network reconstruction in molecular biology.