An introduction to commutative and noncommutative Gröbner bases

Mora, Teo

doi:10.1016/0304-3975(94)90283-6

Cited by 213 publications

(148 citation statements)

References 18 publications

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“…For example, Mora gave an introduction to commutative and noncommutative Gröbner bases, which includes Gröbner bases for Wely algebra (Mora, 1994). As in Insa and Pauer's paper, the rings of coefficients in this paper are general commutative rings, which is the main difference from other existing works.…”

Section: Introductionmentioning

confidence: 95%

On computing Gröbner bases in rings of differential operators

Sun

Wang³

2011

Sci. China Math.

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Insa and Pauer presented a basic theory of Gröbner basis for differential operators with coefficients in a commutative ring in 1998, and a criterion was proposed to determine if a set of differential operators is a Gröbner basis. In this paper, we will give a new criterion such that Insa and Pauer's criterion could be concluded as a special case and one could compute the Gröbner basis more efficiently by this new criterion.

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Section: Introductionmentioning

confidence: 95%

On computing Gröbner bases in rings of differential operators

Sun

Wang³

2011

Sci. China Math.

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“…We call an ordering, having such properties, a monomial ordering on K x, y . See e. g. [8,4,9] for the Gröbner bases theory for K x, y . Lemma 3.…”

Section: Preliminariesmentioning

confidence: 99%

On Two-Generated Non-commutative Algebras Subject to the Affine Relation

Levandovskyy

Koutschan

Motsak

2011

Lecture Notes in Computer Science

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Abstract. We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + αx + βy + γ for q ∈ K * and α, β, γ ∈ K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y m · x n in terms of standard monomials x i y j for many algebras of the considered type. Such formulas are used in e. g. establishing formulas of binomial type and in an implementation of non-commutative multiplication in a computer algebra system. By using the formulas we also study centers and ring-theoretic properties of the non-commutative model algebras.In this paper we study non-commutative algebras in two generators obeying single affine relation. Many operator algebras, coming from different areas of natural sciences, are built from algebras in two generators, see Sect. 2.1 for examples. One of generators, say x, often corresponds to the operator of the multiplication with the function x. Another operator, say y, corresponds to a linear operator, acting on functions in the variable x.In the main Theorem we identify precisely five types of non-isomorphic algebras, which we call model algebras, among them. Despite the fact that many such algebras have been studied in the literature (see e. g. [3, 7, 2, 1], many aspects and properties are too scattered in the existing literature. Another point of this note is to search systematically for closed form of multiplication formulas on monomials. Such closed forms are needed, among other, in computer algebra, where many sophisticated algorithms heavily rely on basic multiplication among monomials. It is not enough to have such formulas just for model algebras, since isomorphisms do not preserve monomials but turn them into polynomials. It turned out, that there are still several cases, where we were not able to derive

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“…(See also [27] for a summary.) For the theory of Gröbner bases, we refer to [5,6], for its noncommutative extension to [3,21]. In this section, we want to define the algebra of integro-differential operators directly in terms of their normal forms.…”

Section: The Algebra Of Integro-differential Operatorsmentioning

confidence: 99%

A skew polynomial approach to integro-differential operators

Regensburger

Rosenkranz

Middeke

2009

Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

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We construct the algebra of integro-differential operators over an ordinary integro-differential algebra directly in terms of normal forms. In the case of polynomial coefficients, we use skew polynomials for defining the integro-differential Weyl algebra as a natural extension of the classical Weyl algebra in one variable. Its normal forms, algebraic properties and its relation to the localization of differential operators are studied. Fixing the integration constant, we regain the integro-differential operators with polynomial coefficients.

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An introduction to commutative and noncommutative Gröbner bases

Cited by 213 publications

References 18 publications

On computing Gröbner bases in rings of differential operators

On computing Gröbner bases in rings of differential operators

On Two-Generated Non-commutative Algebras Subject to the Affine Relation

A skew polynomial approach to integro-differential operators

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