The goal is to cover at least the first four chapters of the book Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Third Edition, by David Cox, John Little, and Donal O'Shea, Springer, New York, 2007. Note: the authors of the book have a web page with errata, software links, etc. Here's some history .
Some of the assignments in this course will involve the use of computer algebra systems. No previous experience with computer programming is assumed, but I expect that you are able and willing to familiarize yourself with the use of the program of your choice.
You are welcome to use a general-purpose program such as Mathematica or Maple (which can do algebra, calculus, graphics, and so on). If you prefer, you may also use Singular, Macaulay2, CoCoA or Sage. These free software systems are explicitly designed to support computations in algebraic geometry and commutative algebra. All these systems are available for most platforms (Unix, Linux, Mac OS X, Windows, etc.).
The instructor's software of choice for this course will be Macaulay2.
If you decide to use Macaulay2, you might want to consult a chapter by Bernd Sturmfels from a book on Macaulay2. Information on how to use Mathematica/Maple for computations with Gröbner bases may be found in Appendix C of the textbook. (Note: Maple packages tend to be rather slow in comparison with a dedicated system such as Macaulay2.
p.s. ideally... a follow-up to this course would be based on this recent preprint: Introduction to non-linear algebra and applications