Computational Commutative Algebra 1

This is a book about Gröbner bases and their applications. It contains 3 chapters, 20 sections, 44 tutorials, 165 exercises, and numerous further amusements. It is going to help you bridge the gap between theoretical computer algebra and actual computation. We hope you will have as much fun reading it as the authors had writing it! From the reviews: "This is one of the most refreshing mathematical books I have ever held in my hands. This is academic teaching at its best; if I had not seen it, I would not have believed that it could be done so well." (Hans Stetter, IMN - Internationale Mathematische Nachrichten 2003) "Every paragraph of the book shows how much the authors have enjoyed translating into printed matter the outcome of a long, large, deep and personal relation with computationally oriented commutative algebra. And the result is a non-standard, elementary and self-contained introduction to the theory of Gröbner bases and its applications." (Laureano González-Vega and Tomás Recio, ACM SIGSAM Bulletin 2004) "The style of this book merits a comment. Each section begins with a quotation and an overview in which "Italian imagination overtakes German rigor". These introductions and the following main bodies of each section are well written, engaging and often amusing. The book is a pleasure to read." (John Little, Mathematical Reviews 2001)

This is a book about Gröbner bases and their applications. It contains 3 chapters, 20 sections, 44 tutorials, 165 exercises, and numerous further amusements.

It is going to help you bridge the gap between theoretical computer algebra and actual computation. We hope you will have as much fun reading it as the authors had writing it!

From the reviews:

This is one of the most refreshing mathematical books I have ever held in my hands. This is academic teaching at its best; if I had not seen it, I would not have believed that it could be done so well. (Hans Stetter, IMN - Internationale Mathematische Nachrichten 2003)

Every paragraph of the book shows how much the authors have enjoyed translating into printed matter the outcome of a long, large, deep and personal relation with computationally oriented commutative algebra. And the result is a non-standard, elementary and self-contained introduction to the theory of Gröbner bases and its applications. (Laureano González-Vega and Tomás Recio, ACM SIGSAM Bulletin 2004)

The style of this book merits a comment. Each section begins with a quotation and an overview in which Italian imagination overtakes German rigor. These introductions and the following main bodies of each section are well written, engaging and often amusing. The book is a pleasure to read. (John Little, Mathematical Reviews 2001)

Hofstadter’s Law: It always takes longer than you think it will take, even if you take into account Hofstadter’s Law. (Douglas R. Hofstadter) Dear Reader, what you are holding in your hands now is for youabook. But for us, for our families and friends, it has been known as the book over the last three years. Three years of intense work just to ?ll three centimeters of your bookshelf! This amounts to about one centimeter per year, or roughly two-?fths of an inch per year if you are non-metric. Clearly we had ample opportunity to experience the full force of Hofstadter’s Law. Writing a book about Computational Commutative Algebra is not - like computing a Gr ̈ obner basis: you need unshakeable faith to believe that the project will ever end; likewise, you must trust in the Noetherianity of polynomial rings to believe that Buchberger’s Algorithm will ever terminate. Naturally, we hope that the ?nal result proves our e?orts worthwhile. This is a book for learning, teaching, reading, and, most of all, enjoying the topic at hand. This introduction to polynomial rings, Gröbner bases and applications bridges the gap in the literature between theory and actual computation. It details numerous applications, covering fields as disparate as algebraic geometry and financial markets. To aid in a full understanding of these applications, more than 40 tutorials illustrate how the theory can be used. The book also includes many exercises, both theoretical and practical. Front Matter....Pages i-xxiii Foundations....Pages 15-83 Gröbner Bases....Pages 85-144 First Applications....Pages 145-273 Back Matter....Pages 275-321