On the construction of Gröbner bases using syzygies

Möller, H. Michael

doi:10.1016/s0747-7171(88)80052-x

Cited by 62 publications

(57 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The variables a are considered as parameters. In this case R[a] can be embedded in the quotient field R(a) and we do not need the general algorithm for computing Gröbner basis over rings (Möller, 1988), but only the classical (Buchberger, 1965(Buchberger, , 1985Gebauer and Möller, 1987;Gianni and Mora, 1987) algorithm over fields.…”

Section: When Does a Gröbner Basis G Of An Ideal I ⊆ R[x] Map To A Grmentioning

confidence: 99%

A New Algorithm for Discussing Gröbner Bases with Parameters

Montes

2002

Journal of Symbolic Computation

105

View full text Add to dashboard Cite

Let F be a set of polynomials in the variables x = x 1 , . . . , xn with coefficients in R[a], where R is a UFD and a = a 1 , . . . , am a set of parameters. In this paper we present a new algorithm for discussing Gröbner bases with parameters. The algorithm obtains all the cases over the parameters leading to different reduced Gröbner basis, when the parameters in F are substituted in an extension field K of R. This new algorithm improves Weispfenning's comprehensive Gröbner basis CGB algorithm, obtaining a reduced complete set of compatible and disjoint cases. A final improvement determines the minimal singular variety outside of which the Gröbner basis of the generic case specializes properly. These constructive methods provide a very satisfactory discussion and rich geometrical interpretation in the applications.

show abstract

Section: When Does a Gröbner Basis G Of An Ideal I ⊆ R[x] Map To A Grmentioning

confidence: 99%

A New Algorithm for Discussing Gröbner Bases with Parameters

Montes

2002

Journal of Symbolic Computation

105

View full text Add to dashboard Cite

show abstract

“…• Syzygy method proposed by a number of researchers including Shtokhamer, Trinks, Zacharias, Schaller and Möller which works for polynomial ideals over Noetherian rings in which certain kinds of syzygies can be solved (see [14,1], etc. ).…”

Section: Related Work: Generalization Of Buchberger's Algorithm For Pmentioning

confidence: 99%

An Algorithm for Computing a Gröbner Basis of a Polynomial Ideal over a Ring with Zero Divisors

Kapur

Cai

2009

Math.Comput.Sci.

View full text Add to dashboard Cite

An algorithm for computing a Gröbner basis of an ideal of polynomials whose coefficients are taken from a ring with zero divisors, is presented; such rings include Z n and Z n [i], where n is not a prime number. The algorithm is patterned after (1) Buchberger's algorithm for computing a Gröbner basis of a polynomial ideal whose coefficients are from a field and (2) its extension developed by Kandri-Rody and Kapur when the coefficients appearing in the polynomials are from a Euclidean domain. The algorithm works as Buchberger's algorithm when a polynomial ideal is over a field and as KandriRody-Kapur's algorithm when a polynomial ideal is over a Euclidean domain. The proposed algorithm and the related technical development are quite different from a general framework of reduction rings proposed by Buchberger in 1984 and generalized later by Stifter to handle reduction rings with zero divisors. These different approaches are contrasted along with the obvious approach where for instance, in the case of Z n , *

show abstract

“…An introduction to Gröbner bases can be found in [11]. It has, for example, been remarked in [17] = (1, . .…”

Section: Corollary 44 If H Is An H-basis Then the Reduced Polynomimentioning

confidence: 99%

Gröbner bases, H–bases and interpolation

Sauer

2000

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The paper is concerned with a construction for H-bases of polynomial ideals without relying on term orders. The main ingredient is a homogeneous reduction algorithm which orthogonalizes leading terms instead of completely canceling them. This allows for an extension of Buchberger's algorithm to construct these H-bases algorithmically. In addition, the close connection of this approach to minimal degree interpolation, and in particular to the least interpolation scheme due to de Boor and Ron, is pointed out.

show abstract

On the construction of Gröbner bases using syzygies

Cited by 62 publications

References 6 publications

A New Algorithm for Discussing Gröbner Bases with Parameters

A New Algorithm for Discussing Gröbner Bases with Parameters

An Algorithm for Computing a Gröbner Basis of a Polynomial Ideal over a Ring with Zero Divisors

Gröbner bases, H–bases and interpolation

Contact Info

Product

Resources

About