# Gröbner bases - a computational approach to commutative algebra

@inproceedings{Becker1993GrbnerB, title={Gr{\"o}bner bases - a computational approach to commutative algebra}, author={Thomas Becker and Volker Weispfenning and Heinz Kredel}, booktitle={Graduate texts in mathematics}, year={1993} }

1: Commutative Rings with Unity. 2: Polynomial Rings. 3: Vector Spaces and Modules. 4: Orders and Abstract Reduction Relations. 5: Gr bner Bases. 6: First Applications of Gr bner Bases. 7: Field Extensions and the Hilbert Nullstellensatz. 8: Decomposition, Radical, and Zeroes of Ideals. 9: Linear Algebra in Residue Class Rings. 10: Variations on Gr bner Bases.

## 1,115 Citations

Grobner Bases in Commutative Algebra

- Mathematics
- 2011

Polynomial rings and ideals Grobner bases First applications Grobner bases for modules Grobner bases of toric ideals Selected applications in commutative algebra and combinatorics Bibliography Index

On the Finiteness of Gröbner Bases Computation in Quotients of the Free Algebra

- MathematicsApplicable Algebra in Engineering, Communication and Computing
- 2001

A Gröbner bases theory is proposed for the authors' factor algebras, of particular interest for one-sided ideals, and a few applications are shown, e.g. how to compute (one-sided) syzygy modules.

Gröbner Bases: An Introduction

- MathematicsICALP
- 1992

The method (theory plus algorithms) of Gröbner bases provides a uniform approach to solving a wide range of problems expressed in terms of sets of multivariate polynomials. x y 2y z z 0 y 2 x 2 z x z…

Gröbner bases in function rings - A guide for introducing reduction relations to algebraic structures

- MathematicsJ. Symb. Comput.
- 2005

Gröbner Bases over Commutative Rings and Applications to Coding Theory

- Mathematics, Computer ScienceGröbner Bases, Coding, and Cryptography
- 2009

The main algorithms that may be implemented to compute Grobner and (in the case of a chain ring) Szekeres-like bases and a number of decoding algorithms for alternant codes over commutative finite chain rings are discussed.

GROEBNER BASES FOR A CLASS OF IDEALS IN COMMUTATIVE POLYNOMIAL RINGS

- Mathematics
- 2010

We construct reduced Groebner bases for a certain class of ideals in commutative polynomial rings. A subclass of these ideals corresponds to the generalized Reed-Muller codes when considered in the…

Analogs of Gröbner Bases in Polynomial Rings over a Ring

- MathematicsJ. Symb. Comput.
- 1996

Abstract In this paper we will define analogs of Grobner bases for R -subalgebras and their ideals in a polynomial ring R [ x 1 ,..., x n ] where R is a noetherian integral domain with multiplicative…

Grobner Bases Algorithm

- Computer Science, Mathematics
- 1995

The Gröbner basis technique is applied to solve systems of polynomial equations in several variables and this technical report investigates this application.

On the Complexity of Computing Gröbner Bases in Characteristic 2

- Mathematics, Computer ScienceInf. Process. Lett.
- 1994

D-Bases for Polynomial Ideals over Commutative Noetherian Rings

- MathematicsRTA
- 1997

A completion-like procedure for constructing D-bases for polynomial ideals over commutative Noetherian rings with unit under certain assumptions about the strategy that controls the application of the transition rules.