Ideals, Varieties, and AlgorithmsFifth Edition, 2025
Springer
Ideals, Varieties, and AlgorithmsThis book discusses systems of polynomial equations ("ideals"), their solutions ("varieties"), and how these objects can be manipulated ("algorithms"). As such, it is an introduction to computational algebraic geometry and commutative algebra at the undergraduate level.
The fifth edition of IVA fixes the typographical errors in the fourth edition and includes new examples, exercises, and projects that provide brief introductions to toric varieties, monomial curves, numerical algebraic geometry, Galois theory, and permanents. The new material was added at the end of existing sections to ensure that the numbering of the results and the exercises in the fifth edition is fully compatible with that in the fourth edition.
In 2016, Ideals, Varieties, and Algorithms was awarded the Leroy P. Steele Prize for Mathematical Exposition by the American Mathematical Society. The article The Story of Ideals, Varieties, and Algorithms tells how the book came to be written.
The fifth edition was originally published in 2025. Typos known as of August 2025 are available in pdf .
The fourth edition was originally published in 2015. A corrected publication appeared in 2018.
Lists of typographical errors are available for the third edition. There are two lists, depending on which printing you have. Because of the fourth edition, the typo list for the third edition is no longer being updated.
To find out which printing you have, check the second line from the bottom on the copyright page; the last digit displayed is the printing number. If you have the first printing, you will need to download both lists; if you have the second printing, you only need the second.
A complete solutions manual for the 4th and 5th editions of Ideals, Varieties, and Algorithms has been written up by David Cox and Ying Li of St. Francis University.
The solutions are not posted here because having open access to the solutions would diminish the value of the text. Ideals, Varieties, and Algorithms is a book where you learn by doing. If you are teaching from Ideals, Varieties, and Algorithms or are studying the book on your own, you may obtain a pdf copy of the solutions by sending email to donalboshea@gmail.com.
The book describes the computer algebra systems Maple, Mathematica and SageMath in some detail. Maple and Mathematica are commercial products, while SageMath is freely available. at
In addition, here are some other computer algebra programs which can do Gröbner basis calculations: Of these, all are free except for Magma, though Magma has a free web-based calculator for small computations..
Click here for the web page for the book Using Algebraic Geometry. This book is an introduction to Gröbner bases and resultants, which are two of the main tools used in computational algebraic geometry and commutative algebra. It also discusses local methods and syzygies, and gives applications to integer programming, polynomial splines and algebraic coding theory. The second edition, published by Springer in 2005, contains several new sections on Gröbner fans of ideals, the Gröbner Walk basis conversion algorithm, and a new chapter on codes from order domains and the Berlekamp-Massey-Sakata decoding algorithm. It is available in both hardcover and paperback.
The catalog entry for Ideals, Varieties, and Algorithms in the Springer on-line catalog contains a brief description of the book and also includes ordering information.
You can contact the authors at the following email addresses:
dacox@amherst.edu jlittle@holycross.edu donalboshea@gmail.com