Intervals, Syzygies, Numerical Gröbner Bases: A Mixed Study (original) (raw)

2006, Lecture Notes in Computer Science

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.

Numerical Grobner Bases and Syzygies: an Interval Approach

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.

Numerical computation of Gröbner bases

Proceedings of CASC2004 ( …, 2004

In this paper we deal with the problem of numerical computation of Gröbner bases of zero-dimensional polynomial systems. It is well known that the computation of a Gröbner basis cannot be generally executed in floating-point arithmetic by a standard approach. This, however, would be highly desirable for practical applications. We present an approach for computing Gröbner bases numerically. It is an elaboration of the idea of a stabilized Gröbner basis computation initially proposed by Hans Stetter. Our implementation of the algorithm based on the presented results is available online.

Gröbner bases computation using syzygies

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

Gröbner Bases and Polynomial Equations

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...

Groebner Bases and an Improvement on Buchberger's Algorithm

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.

Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases

… of the 2009 international symposium on …, 2009

In this paper, we propose an efficient method to solve polynomial systems whose equations are left invariant by the action of a finite group G. The idea is to simultaneously compute a truncated SAGBI-Gröbner bases (a generalisation of Gröbner bases to ideals of subalgebras of polynomial ring) and a Gröbner basis in the invariant ring K[σ1, . . . , σn] where σi is the i-th elementary symmetric polynomial.

Floating-Point Gröbner Basis Computation with Ill-conditionedness Estimation

Lecture Notes in Computer Science, 2008

Computation of Gröbner bases of polynomial systems with coefficients of floating-point numbers has been a serious problem in computer algebra for a long time; the computation often becomes very unstable and people did not know how to remove the instability. Recently, the present authors clarified the origin of instability and presented a method to remove the instability. Unfortunately, the method is very time-consuming and not practical. In this paper, we first investigate the instability much more deeply than in the previous paper, then we give a theoretical analysis of the term cancellation which causes large errors, in various cases. On the basis of this analysis, we propose a practical method for computing the Gröbner bases with coefficients of floating-point numbers. The method utilizes multiple precision floating-point numbers, and it removes the drawbacks of the previous method almost completely. Furthermore, we present a method of estimating the ill-conditionedness of the input system.

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