An Introduction to Gröbner Bases

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
* SoftwareGrobner-Basen werden von Mathematikern und Informatikern zunehmend fur eine breite Palette von Anwendungen genutzt, in denen die algorithmische algebraische Geometrie eine Rolle spielt. Hier werden Grobner-Basen von einem konstruktiven, wenig abstrakten Standpunkt aus behandelt, wobei nur geringe Vorkenntnisse in linearer Algebra und komplexen Zahlen vorausgesetzt werden; zahlreiche Beispiele helfen bei der Durchdringung des Stoffes. Mit einer Ubersicht uber aktuell erhaltliche relevante Softwarepakete. Rings.

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.

Software.

Hints to Some Exercises.

Answers to Exercises.

Bibliography.

Index. Rings, Fields and Ideals, Monomial Ideals, Monomial Orders, Algebraic Sets, Improvements to Buchberger's Algorithms, Applications of Grobner Bases, Hilbert Series, Variations of Grobner Bases, Available Software.

Ralf Fröberg is the author of An Introduction to Gröbner Bases, published by Wiley. 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