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 Groebner 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 Maple (TM), Mathematica (R) 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. Readers who are teaching from Ideals, Varieties, and
Algorithms, or are studying the book on their own, may obtain a
copy of the solutions manual by sending an email to
[email protected]. 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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