The GANITH algebraic geometry toolkit

Abstract

We are building a general-purpose tool for computing and visualizing solutions to systems of algebraic equations. Diverse algorithms exist for this problem and related sub-problems, and we shall incorporate several of them. We are also developing a new set of solution techniques using multi-polynomial resultants and birational maps between arbitrary algebraic sets and hypersurfaces. Our designed tool shall be portable and allow rapid prototyping of new and existing algorithms, in an intelligent blend of algebraic and numeric methods, and real solutions. Complex solutions can also be computed, but visualizing them is difficult. The software shall also take advantage of any parallel hardware that is present. In all, we anticipate improving the state of the art of algebraic equation solving by devising novel techniques, blending existing methods to form new ones, and comparing the relative efficacy of the various methods.

A prototype system implementing some of these goals is built and evolving. It is written in Common Lisp and C, and runs under version 11 of the X window system. The system is portable to any machine that runs vanilla Common Lisp, supports the X window system, and has at least a rudimentary Lisp/C interface. At the present time it can solve systems of up to two equations in any number of variables. Example applications of this are curve and surface display, curve-curve intersections, surface-surface intersections, etc. A graphical user interface allows the display and manipulation of 0D (points), 1D (curves) or 2D (surface) solutions.

Supported in part by NSF grant DMS 88-16286, ARO contract DAAG29-85-C-0018 under Cornell MSI and ONR contract N0001 4-88-K-0402