Discriminants, Resultants, and Multidimensional Determinants

Springer Science & Business Media
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"This book revives and vastly expands the classical theory of resultants and discriminants. Most of the main new results of the book have been published earlier in more than a dozen joint papers of the authors. The book nicely complements these original papers with many examples illustrating both old and new results of the theory."—Mathematical Reviews

"Collecting and extending the fundamental and highly original results of the authors, it presents a unique blend of classical mathematics and very recent developments in algebraic geometry, homological algebra, and combinatorial theory." —Zentralblatt Math

"This book is highly recommended if you want to get into the thick of contemporary algebra, or if you wish to find some interesting problem to work on, whose solution will benefit mankind." —Gian-Carlo Rota, Advanced Book Reviews

"...the book is almost perfectly written, and thus I warmly recommend it not only to scholars but especially to students. The latter do need a text with broader views, which shows that mathematics is not just a sequence of apparently unrelated expositions of new theories, ... but instead a very huge and intricate building whose edification may sometimes experience difficulties ... but eventually progresses steadily." —Bulletin of the American Mathematical Society

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Additional Information

Springer Science & Business Media
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Published on
May 21, 2009
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Mathematics / Algebra / Abstract
Mathematics / Algebra / General
Mathematics / Algebra / Linear
Mathematics / Geometry / Algebraic
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This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).

The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course. It is assumed that the reader has access to a computer algebra system. Appendix C describes features of MapleTM, Mathematica® and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used.

From the reviews of previous editions:

“...The book gives an introduction to Buchberger’s algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. ...The book is well-written. ...The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry.”

—Peter Schenzel, zbMATH, 2007

“I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry.”

—The American Mathematical Monthly

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