Groebner Basis (GB) is a recognized tool for solving problems formulated in terms of systems of polynomial equations in several variables. I have authored a package called CGBlisp about 10 years ago, which implemented Groebner basis and a generalization due to Weispfenning, which is called Comprehensive Groebner Bases (CGB).
Subsequently, I extracted a small portion of CGBLisp and interfaced it to a Computer Algebra System (CAS) called Maxima. The package called Maxima_Grobner is distributed with Maxima, as a contributed package. Here is a link to the Maxima project:
My development site containing the current version of the package can be found here:
Unfortunately, due to some neglect, does not work out of the box with current Maxima distributions. The reason is really simple and easy to correct by a Lisp programmer, but perhaps not by the general audience interested in using the high-level CAS. The fix consists basically in removing the optimization constructs which in the original distribution of Maxima_Grobner were designed to work with CMU Common Lisp, Allegro and GCL dialects of Common Lisp. Today, the Common Lisp used by Maxima is SBCL. SBCL, derived from CMU Common Lisp, appears to handle compilation of Maxima_Grobner well enough without special directives. Short of combing through the code again, I simply removed the optimization directives from the code, loaded it into Maxima and was able to run the demo without problems.
There appears to be renewed interest in CGB. Many interesting problems, both mathematical and applied, can be formulated in terms of systems of polynomial equations with parameters. CGB allow one to answer many questions about the solutions of such systems, shed light on the special parameters involved and allow one to solve examples that would otherwise be intractible. Thus, it should be of value to large audiences.
This book will bring my updated software for computing GB and CGB. This project should take me several months to complete. For now, please be assured that CGBLisp and Maxima_Grobner are fully functional. This is the place to look for further announcements.
As attachments to this page, I included a copy of my published article on CGBLisp, and a copy of Bill Dunn's dissertation, which explains the underpinnings of CGB. They are available to subscribed users.