I did an REU (Research Experience for Undergraduates) this summer and we used the textbook “Ideals, Varieties, and Algorithms” (isbn 978-0-387-35650-1) by David Cox, John Little, and Donal O’Shea. As the title mentions, algorithms are an important tool in the study of varieties via their corresponding algebraic ideals. One of the main tools used in algebraic geometry is what is known as a Groebner basis, which is somewhat akin to a basis set in Linear Algebra. Anyway, I wrote some C++ code which implements a few of the algorithms in the book, but stops short of implementing Buchberger’s algorithm for computing a Groebner basis. The code should be useful as a starting point because it implements a Monomial class and a Polynomial class both with lots of useful methods thus eliminating the drudgery of implementing the book’s algorithms in C++.
The code can be found at: my web site as a gzipped tarball. Please let me know if you find it useful. I developed the code on an Ubuntu Linux machine, but it is generic enough that it should compile on any platform. You will need to read the README to understand the simple input, output format.