Computation of Dimension in Filtered Free Modules by Gröbner Reduction

Fürst, Christoph; Landsmann, Günter

doi:10.1145/2755996.2756680

Cited by 2 publications

(5 citation statements)

References 9 publications

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“…This representaion allows the construction of Gröbner bases for submodules of finitely generated free modules over D out of Gröbner bases for the corresponding modules over O. Cf. also Lemma 3.1 and [15].…”

Section: The Ring Of Difference-differential Operatorsmentioning

confidence: 81%

“…have been shown to be subordinate to this scheme. It has then be proved that reduction concepts which obey the axioms of Gröbner reduction allow the derivation of the dimension polynomial of finitely generated modules [15]. Later it has been shown that concepts like reduction relative to several term orders or relative reduction can be expressed in terms of Gröbner reductions [16].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we leave the tight environment that was dictated by the conception of [15]. The notion of reduction relation in a module is analyzed with regard to maximizing the scope of its models.…”

Section: Introductionmentioning

confidence: 99%

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An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles

Landsmann¹,

Fürst²

2021

AJAM

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Several variants of the classical theory of Gröbner bases can be found in the literature. They come, depending on the structure they operate on, with their own specific peculiarity. Setting up an expedient reduction concept depends on the arithmetic equipment that is provided by the structure in question. Often it is necessary to introduce a term order that can be used for determining the orientation of the reduction, the choice of which might be a delicate task. But there are other situations where a different type of structure might give the appropriate basis for formulating adequate rewrite rules. In this paper we have tried to find a unified concept for dealing with such situations. We develop a global theory of Gröbner bases for modules over a large class of rings. The method is axiomatic in that we demand properties that should be satisfied by a reduction process. Reduction concepts obeying the principles formulated in the axioms are then guaranteed to terminate. The class of rings we consider is large enough to subsume interesting candidates. Among others this class contains rings of differential operators, Ore-algebras and rings of difference-differential operators. The theory is general enough to embrace the well-known classical Gröbner basis concepts of commutative algebra as well as several modern approaches for modules over relevant noncommutative rings. We start with introducing the appropriate axioms step by step, derive consequences from them and end up with the Buchberger Algorithm, that makes it possible to compute a Gröbner basis. At the end of the paper we provide a few examples to illustrate the abstract concepts in concrete situations.

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Section: The Ring Of Difference-differential Operatorsmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

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An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles

Landsmann¹,

Fürst²

2021

AJAM

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“…With the above notation, one can consider the following concept of Gröbner reduction first introduced in [7].…”

Section: Remark 24mentioning

confidence: 99%

“…It is shown in [7,Section 3], that for the bivariate filtration F r,s , using an appropriate choice of term orders ≺ 1 and ≺ 2 it is possible to ensure condition f ∈ F r,s and f −→ h ⇒ h ∈ F r,s . In particular, this property holds with respect to the orders given in [22,Section 3].…”

Section: −C N ) (Ii) Every Z (N)mentioning

confidence: 99%

Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

Fürst

Levin

2017

Math.Comput.Sci.

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In this paper we develop a relative Gröbner basis method for a wide class of filtered modules. Our general setting covers the cases of modules over rings of differential, difference, inversive difference and differencedifferential operators, Weyl algebras and multiparameter twisted Weyl algebras (the last class of rings includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras). In particular, we obtain a Buchbergertype algorithm for constructing relative Gröbner bases of filtered free modules.

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Computation of Dimension in Filtered Free Modules by Gröbner Reduction

Cited by 2 publications

References 9 publications

An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles

An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles

Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

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