cover of book Ideals, Varieties, and Algorithms
Third Edition, 2007
Springer

by

David A. Cox,
Amherst College
John B. Little,
College of the Holy Cross
Don O'Shea,
Mount Holyoke College


Note to Readers (January, 2007):

In the 5th printing of the second edition of IVA, a production error caused plus signs to appear as minus signs on many pages. Hence the 5th printing of the book is defective. Springer has been aware of this problem since November 2005. Owners of the 5th printing should contact Springer about obtaining a corrected copy.

The third edition of IVA is now available.


Contents of Web Page:



What's in the Book

This book is an introduction to computational algebraic geometry and commutative algebra at the undergraduate level. It discusses systems of polynomial equations ("ideals"), their solutions ("varieties"), and how these objects can be manipulated ("algorithms").

The Table of Contents of Ideals, Varieties, and Algorithms gives a more detailed picture of what the book covers.


Computer Packages

On pages 522 and 524 of the third edition (= pages 510 and 512 of the second edition), Appendix C mentions computer packages for Maple and Mathematica. The packages were written for teaching purposes and tend to be very slow. Thus you can use them only on very simple problems.

For Releases 2, 3 and 4 of Maple V, the following versions of the Maple package are available:

The main commands in the Maple package are div_alg, which computes quotients and remainders in the division algorithm, and mxgb, which computes a Gröbner basis together with a matrix showing how to express the Gröbner basis in terms of the original basis.

For Release 5 of Maple V, an expanded version of the Maple package is available. This version includes commands for computing Gröbner bases which give intermediate steps as well as tutorial and reference worksheets. The files involved are:

The Maple package can be used without either of the worksheets. The tutorial and reference worksheets, on the other hand, require that you also download the package. The tutorial worksheet discusses the various commands in the package and gives examples of how they work, while the reference worksheet gives the precise syntax of each command in the package.

WARNING: The IVA Maple package for Release 5 of Maple V is incompatible with Maple 6 and Maple 7. For Maple 6 and 7, you should download the following files:

As far as we know, these work with more recent releases of Maple, including Maple 9 and 10.

For Mathematica, the following versions of the Mathematica package are available:

WARNING: The Mathematica 5 versions of the Package and Notebook do not work with Mathematica 6.

When using these Mathematica packages and notebooks, the following should be kept in mind:

In general, if you have difficulties with either of the packages or suggestions for improvements, please let us know! (Thanks to David Vella and his students at Skidmore for letting me know that the 4.0 version of the package doesn't work with Mathematica 5.0. They also suggested the solution implemented in the 5.0 version of the package.)


Typographical Errors

Third Edition

Lists of typographical errors are available for the third edition. There are two lists, depending on which printing you have.

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.

Second Edition

Lists of typographical errors are available for the second edition. There is a separate list for each printing. 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.

First Edition

Lists of typographical errors are also available for the first edition. There is a separate list for each printing. To find out which printing you have, check the third line from the bottom on the copyright page; the last digit displayed is the printing number.


Solutions to Selected Exercises

Solutions to various exercises from the first four chapters have been written up by Fred Woodward under the direction of Michael Singer of North Carolina State University. We are grateful to Fred and Michael for allowing us to use their solutions.

Solutions to selected exercises from the whole book have been written up by Hyeyoun Chung for a seminar supervised by Steven Kleiman of MIT. We are grateful to Hyeyoun and Steve for allowing us to use their solutions.

The solutions are not posted here because some instructors don't want their students to have access to solutions. However, if you are teaching from Ideals, Varieties, and Algorithms or are studying the book on your own, you may obtain a copy of the solutions by sending email to dac@cs.amherst.edu. When requesting a copy, please let us know whether you would prefer the solutions sent in a LaTeX file, a postscript file, or a pdf file. Hardcopy is also available if for some reason the electronic versions aren't suitable.


Note to Maple Users

Appendix C of Ideals, Varieties, and Algorithms includes a section that explains how Maple can be used for Gröbner basis and related computations. The first edition of the book describes the now obsolete grobner package, while the second and third editions describe the Groebner package for Math 9 and 9.5 respectively.

The current version of Maple is Maple 13, which has an enhanced version of Groebner package as well as the new PolynomialIdeals package. An updated version of the Maple section of Appendix C that describes these packages is available as either pdf or postscript .


Computer Algebra Programs

The book describes the computer algebra systems AXIOM, Maple, Mathematica and REDUCE in some detail. In addition, here are some other computer algebra programs which can do Gröbner basis calculations:

Of these, all are free except for Magma.


Interesting Links


Using Algebraic Geometry

Click here for the web page for our 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 was published by Springer in the summer of 2005. It is available in both hardcover and paperback.



Contacting Springer-Verlag

The catalog entry for Ideals, Varieties, and Algorithms in the Springer-Verlag on-line catalog contains a brief description of the book and also includes ordering information.


Contacting the Authors

You can contact the authors at the following email addresses:

dac@math.amherst.edu
little@math.holycross.edu
doshea@mhc.mtholyoke.edu