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

Günter Landsmann; Christoph Fürst

doi:doi:10.11648/j.ajam.20210904.14

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

Günter Landsmann, Christoph Fürst

Published in American Journal of Applied Mathematics (Volume 9, Issue 4)

Received: 28 May 2021 Accepted: 26 June 2021 Published: 2 August 2021

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Abstract

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.

Published in	American Journal of Applied Mathematics (Volume 9, Issue 4)
DOI	10.11648/j.ajam.20210904.14
Page(s)	123-140
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Gröbner Bases, Reduction Relations, Rings of Differential Operators, Differential Algebra

References

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Günter Landsmann, Christoph Fürst. (2021). An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles. American Journal of Applied Mathematics, 9(4), 123-140. https://doi.org/10.11648/j.ajam.20210904.14

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ACS Style

Günter Landsmann; Christoph Fürst. An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles. Am. J. Appl. Math. 2021, 9(4), 123-140. doi: 10.11648/j.ajam.20210904.14

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AMA Style

Günter Landsmann, Christoph Fürst. An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles. Am J Appl Math. 2021;9(4):123-140. doi: 10.11648/j.ajam.20210904.14

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@article{10.11648/j.ajam.20210904.14,
  author = {Günter Landsmann and Christoph Fürst},
  title = {An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles},
  journal = {American Journal of Applied Mathematics},
  volume = {9},
  number = {4},
  pages = {123-140},
  doi = {10.11648/j.ajam.20210904.14},
  url = {https://doi.org/10.11648/j.ajam.20210904.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210904.14},
  abstract = {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.},
 year = {2021}
}

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TY  - JOUR
T1  - An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles
AU  - Günter Landsmann
AU  - Christoph Fürst
Y1  - 2021/08/02
PY  - 2021
N1  - https://doi.org/10.11648/j.ajam.20210904.14
DO  - 10.11648/j.ajam.20210904.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 123
EP  - 140
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20210904.14
AB  - 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.
VL  - 9
IS  - 4
ER  -

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Author Information

Günter Landsmann

Research Institute for Symbolic Computation – RISC Linz, Johannes Kepler University, Linz, Austria
Christoph Fürst

Research Institute for Symbolic Computation – RISC Linz, Johannes Kepler University, Linz, Austria

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APA Style

Günter Landsmann, Christoph Fürst. (2021). An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles. American Journal of Applied Mathematics, 9(4), 123-140. https://doi.org/10.11648/j.ajam.20210904.14

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ACS Style

Günter Landsmann; Christoph Fürst. An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles. Am. J. Appl. Math. 2021, 9(4), 123-140. doi: 10.11648/j.ajam.20210904.14

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AMA Style

Günter Landsmann, Christoph Fürst. An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles. Am J Appl Math. 2021;9(4):123-140. doi: 10.11648/j.ajam.20210904.14

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@article{10.11648/j.ajam.20210904.14,
  author = {Günter Landsmann and Christoph Fürst},
  title = {An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles},
  journal = {American Journal of Applied Mathematics},
  volume = {9},
  number = {4},
  pages = {123-140},
  doi = {10.11648/j.ajam.20210904.14},
  url = {https://doi.org/10.11648/j.ajam.20210904.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210904.14},
  abstract = {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.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles
AU  - Günter Landsmann
AU  - Christoph Fürst
Y1  - 2021/08/02
PY  - 2021
N1  - https://doi.org/10.11648/j.ajam.20210904.14
DO  - 10.11648/j.ajam.20210904.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 123
EP  - 140
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20210904.14
AB  - 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.
VL  - 9
IS  - 4
ER  -

Copy | Download