Book contents
- Frontmatter
- Contents
- Preface
- Programme Committee
- Tutorials
- Introduction to Gröbner Bases
- Gröbner Bases, Symbolic Summation and Symbolic Integration
- Gröbner Bases and Invariant Theory
- A Tutorial on Generic Initial Ideals
- Gröbner Bases and Algebraic Geometry
- Gröbner Bases and Integer Programming
- Gröbner Bases and Numerical Analysis
- Gröbner Bases and Statistics
- Gröbner Bases and Coding Theory
- Janet Bases for Symmetry Groups
- Gröbner Bases in Partial Differential Equations
- Gröbner Bases and Hypergeometric Functions
- Introduction to Noncommutative Gröbner Bases Theory
- Gröbner Bases Applied to Geometric Theorem Proving and Discovering
- Research Papers
- Appendix An Algorithmic Criterion for the Solvability of a System of Algebraic Equations (translated by Michael Abramson and Robert Lumbert)
- Index of Tutorials
Gröbner Bases, Symbolic Summation and Symbolic Integration
Published online by Cambridge University Press: 05 July 2011
- Frontmatter
- Contents
- Preface
- Programme Committee
- Tutorials
- Introduction to Gröbner Bases
- Gröbner Bases, Symbolic Summation and Symbolic Integration
- Gröbner Bases and Invariant Theory
- A Tutorial on Generic Initial Ideals
- Gröbner Bases and Algebraic Geometry
- Gröbner Bases and Integer Programming
- Gröbner Bases and Numerical Analysis
- Gröbner Bases and Statistics
- Gröbner Bases and Coding Theory
- Janet Bases for Symmetry Groups
- Gröbner Bases in Partial Differential Equations
- Gröbner Bases and Hypergeometric Functions
- Introduction to Noncommutative Gröbner Bases Theory
- Gröbner Bases Applied to Geometric Theorem Proving and Discovering
- Research Papers
- 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
The treatment of combinatorial expressions and special functions by linear operators is amenable to Gröbner basis methods. In this tutorial, we illustrate the applications of Gröbner bases to symbolic summation and integration.
Introduction
In the late 1960's, Risch (1969, 1970) developed an algorithm for symbolic indefinite integration. The approach followed there consists in computing a tower of differential extensions in order to determine if an indefinite integral can be expressed in terms of elementary functions. Risch's algorithm became very popular and is now at the heart of the integration routines of many computer algebra systems. In the early 1980's, Karr (1981, 1985) appealed to similar ideas, namely difference extensions, in order to develop an algorithm for symbolic indefinite summation. Despite its indisputable algorithmic interest, Karr's algorithm has unfortunately not received as much attention as it deserves yet, due to its complexity and the difficulty to implement it.
In the early 1990's, Zeilberger (1990b) initiated a different approach to symbolic summation and integration. As opposed to the approach by differential or difference extensions, Zeilberger studies the action of algebras of differential or difference linear operators in order to compute special operators that determine the sum or integral under consideration.
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- Gröbner Bases and Applications , pp. 32 - 60Publisher: Cambridge University PressPrint publication year: 1998
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