Book contents
- Frontmatter
- Contents
- Preface
- Programme Committee
- Tutorials
- Research Papers
- The Fractal Walk
- Gröbner Bases Property on Elimination Ideal in the Noncommutative Case
- 17 The CoCoA 3 Framework for a Family of Buchberger-like Algorithms
- 18 Newton Identities in the Multivariate Case: Pham Systems
- 19 Gröbner Bases in Rings of Differential Operators
- 20 Canonical Curves and the Petri Scheme
- 21 The Buchberger Algorithm as a Tool for Ideal Theory of Polynomial Rings in Constructive Mathematics
- 22 Gröbner Bases in Non-Commutative Reduction Rings
- 23 Effective Algorithms for Intrinsically Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field
- 24 De Nugis Groebnerialium 1: Eagon, Northcott, Gröbner
- 25 An application of Gröbner Bases to the Decomposition of Rational Mappings
- 26 On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings
- 27 Full Factorial Designs and Distracted Fractions
- 28 Polynomial interpolation of Minimal Degree and Gröbner Bases
- 29 Inversion of Birational Maps with Gröbner Bases
- 30 Reverse Lexicographic Initial Ideals of Generic Ideals are Finitely Generated
- 31 Parallel Computation and Gröbner Bases: An Application for Converting Bases with the Gröbner Walk
- Appendix An Algorithmic Criterion for the Solvability of a System of Algebraic Equations (translated by Michael Abramson and Robert Lumbert)
- Index of Tutorials
23 - Effective Algorithms for Intrinsically Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- Preface
- Programme Committee
- Tutorials
- Research Papers
- The Fractal Walk
- Gröbner Bases Property on Elimination Ideal in the Noncommutative Case
- 17 The CoCoA 3 Framework for a Family of Buchberger-like Algorithms
- 18 Newton Identities in the Multivariate Case: Pham Systems
- 19 Gröbner Bases in Rings of Differential Operators
- 20 Canonical Curves and the Petri Scheme
- 21 The Buchberger Algorithm as a Tool for Ideal Theory of Polynomial Rings in Constructive Mathematics
- 22 Gröbner Bases in Non-Commutative Reduction Rings
- 23 Effective Algorithms for Intrinsically Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field
- 24 De Nugis Groebnerialium 1: Eagon, Northcott, Gröbner
- 25 An application of Gröbner Bases to the Decomposition of Rational Mappings
- 26 On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings
- 27 Full Factorial Designs and Distracted Fractions
- 28 Polynomial interpolation of Minimal Degree and Gröbner Bases
- 29 Inversion of Birational Maps with Gröbner Bases
- 30 Reverse Lexicographic Initial Ideals of Generic Ideals are Finitely Generated
- 31 Parallel Computation and Gröbner Bases: An Application for Converting Bases with the Gröbner Walk
- Appendix An Algorithmic Criterion for the Solvability of a System of Algebraic Equations (translated by Michael Abramson and Robert Lumbert)
- Index of Tutorials
Summary
Abstract
Gröbner bases have been an important tool for performing computations in polynomial rings since Buchberger first presented an algorithm to construct them in 1965. A generalized type of Gröbner basis was developed in the context of a valuation ring by Sweedler in 1988. The purpose of this paper is to concretize Sweedler's theory to the context of a k-subalgebra of a polynomial ring k[x1,…,xn], where k is a field and to present effective additional algorithms necessary for intrinsically constructing objects that we shall call SAGBI-Gröbner bases.
Introduction
SAGBI-Gröbner bases are simply the counterparts in ideals of subalgebras of polynomial rings to Gröbner bases of ideals of the full polynomial ring. Our intrinsic development of SAGBI-Gröbner theory will mimic that of ordinary Gröbner bases, an approach that has a certain aesthetic appeal. In addition, this parallel development provides another point of view from which to study the mechanics of Gröbner basis theory, which was first presented in an algorithmic fashion by Buchberger (1965) (see also Buchberger (1985)).
We assume little experience with Gröbner bases on the part of the reader; an overview of the subject may be found in the texts of Adams and Loustaunau (1994), Becker and Weispfennig (1993), and Cox, Little, and O'Shea (1992).
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- Gröbner Bases and Applications , pp. 421 - 433Publisher: Cambridge University PressPrint publication year: 1998
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