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An Introduction to Gröbner Bases

An Introduction to Gröbner Bases

Ralf Fröberg

ISBN: 978-0-470-86620-7 November 2000 190 Pages




As algebra becomes more widely used in a variety of applications and computers are developed to allow efficient calculations in the field, so there becomes a need for new techniques to further this area of research. Gröbner Bases is one topic which has recently become a very popular and important area of modern algebra. This book provides a concrete introduction to commutative algebra through Gröbner Bases. The inclusion of exercises, lists of further reading and related literature make this a practical approach to introducing Gröbner Bases. The author presents new concepts and results of recent research in the area allowing students and researchers in technology, computer science and mathematics to gain a basic understanding of the technique. A first course in algebra is the only prior knowledge required for this introduction. Chapter titles include:
* Monomial ldeas
* Gröbner Bases
* Algebraic Sets
* Solving Systems of Polynomial Equations
* Applications of Gröbner Bases
* Homogeneous Algebra
* Hilbert Series
* Variations of Gröbner Bases
* Improvements to Buchberger's Algorithms
* Software

Fields, and Ideals.

Monomial Ideals.

Gröbner Bases.

Algebraic Sets.

Primary Decomposition of Ideals.

Solving Systems of Polynomial Equations.

Applications of Gröbner Bases.

Homogeneous Algebras.

Projective Varieties.

The Associated Graded Ring.

Hilbert Series.

Variations of Gröbner Bases.

Improvements to Buchberger's Algorithm.


Hints to Some Exercises.

Answers to Exercises.