As the primary tool for doing explicit computations in polynomial rings in many variables, Gröbner bases are an important component of all computer algebra systems. They are also important in computational commutative algebra and algebraic geometry. This book provides a leisurely and fairly comprehensive introduction to Gröbner bases and their applications. Adams and Loustaunau cover the following topics: the theory and construction of Gröbner bases for polynomials with coefficients in a field, applications of Gröbner bases to computational problems involving rings of polynomials in many variables, a method for computing syzygy modules and Gröbner bases in modules, and the theory of Gröbner bases for polynomials with coefficients in rings. With over 120 worked examples and 200 exercises, this book is aimed at advanced undergraduate and graduate students. It would be suitable as a supplement to a course in commutative algebra or as a textbook for a course in computer algebra or computational commutative algebra. This book would also be appropriate for students of computer science and engineering who have some acquaintance with modern algebra.

A very carefully crafted introduction to the theory and some of the applications of Grobner bases ... contains a wealth of illustrative examples and a wide variety of useful exercises, the discussion is everywhere well-motivated, and further developments and important issues are well sign-posted ... has many solid virtues and is an ideal text for beginners in the subject ... certainly an excellent text. --Bulletin of the London Mathematical Society As the primary tool for doing explicit computations in polynomial rings in many variables, Grobner bases are an important component of all computer algebra systems. They are also important in computational commutative algebra and algebraic geometry. This book provides a leisurely and fairly comprehensive introduction to Grobner bases and their applications. Adams and Loustaunau cover the following topics: the theory and construction of Grobner bases for polynomials with coefficients in a field, applications of Grobner bases to computational problems involving rings of polynomials in many variables, a method for computing syzygy modules and Grobner bases in modules, and the theory of Grobner bases for polynomials with coefficients in rings. With over 120 worked-out examples and 200 exercises, this book is aimed at advanced undergraduate and graduate students. It would be suitable as a supplement to a course in commutative algebra or as a textbook for a course in computer algebra or computational commutative algebra. This book would also be appropriate for students of computer science and engineering who have some acquaintance with modern algebra.

Grobner-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.

Comprehensive account of theory and applications of Gröbner bases, co-edited by the subject's inventor.

This book provides a concise yet comprehensive and self-contained introduction to Grobner basis theory and its applications to various current research topics in commutative algebra. It especially aims to help young researchers become acquainted with fundamental tools and techniques related to Grobner bases which are used in commutative algebra and to arouse their interest in exploring further topics such as toric rings, Koszul and Rees algebras, determinantal ideal theory, binomial edge ideals, and their applications to statistics. The book can be used for graduate courses and self-study. More than 100 problems will help the readers to better understand the main theoretical results and will inspire them to further investigate the topics studied in this book.

1. Preliminaries. 1.1. Presenting algebras by relations. 1.2. S-graded algebras and modules. 1.3. [symbol]-filtered algebras and modules -- 2. The [symbol]-leading homogeneous algebra A[symbol]. 2.1. Recognizing A via G[symbol](A): part 1. 2.2. Recognizing A via G[symbol](A): part 2. 2.3. The [symbol-graded isomorphism A[symbol](A). 2.4. Recognizing A via A[symbol] -- 3. Grobner bases: conception and construction. 3.1. Monomial ordering and admissible system. 3.2. Division algorithm and Grobner basis. 3.3. Grobner bases and normal elements. 3.4. Grobner bases w.r.t. skew multiplicative K-bases. 3.5. Grobner bases in K[symbol] and KQ. 3.6. (De)homogenized Grobner bases. 3.7. dh-closed homogeneous Grobner bases -- 4. Grobner basis theory meets PBW theory. 4.1. [symbol]-standard basis [symbol]-PBW isomorphism. 4.2. Realizing [symbol]-PBW isomorphism by Grobner basis. 4.3. Classical PBW K-bases vs Grobner bases. 4.4. Solvable polynomial algebras revisited -- 5. Using A[symbol] in terms of Grobner bases. 5.1. The working strategy. 5.2. Ufnarovski graph. 5.3. Determination of Gelfand-Kirillov Dimension. 5.4. Recognizing Noetherianity. 5.5. Recognizing (semi- )primeness and PI-property. 5.6. Anick's resolution over monomial algebras. 5.7. Recognizing finiteness of global dimension. 5.8. Determination of Hilbert series -- 6. Recognizing (non- )homogeneous p-Koszulity via A[symbol]. 6.1. (Non- )homogeneous p-Koszul algebras. 6.2. Anick's resolution and homogeneous p-Koszulity. 6.3. Working in terms of Grobner bases -- 7. A study of Rees algebra by Grobner bases. 7.1. Defining [symbol] by [symbol]. 7.2. Defining [symbol] by [symbol]. 7.3. Recognizing structural properties of [symbol] via [symbol]. 7.4. An application to regular central extensions. 7.5. Algebras defined by dh-closed homogeneous Grobner bases -- 8. Looking for more Grobner bases. 8.1. Lifting (finite) Grobner bases from O[symbol]. 8.2. Lifting (finite) Grobner bases from a class of algebras. 8.3. New examples of Grobner basis theory. 8.4. Skew 2-nomial algebras. 8.5. Almost skew 2-nomial algebras