Next generation computer algebra systems AXIOM and the scratchpad concept: applications to research in algebra

Abstract

One way in which mathematicians deal with infinite amounts of data is symbolic representation. A simple example is the quadratic equation x = −b±√b2−4ac 2a, a formula which uses symbolic representation to describe the solutions to an infinite class of equations. Most computer algebra systems can deal with polynomials with symbolic coefficients, but what if symbolic exponents are called for (e.g., 1+t i)? What if symbolic limits on summations are also called for (e.g., 1+t+...+t i = � j tj)? The “Scratchpad Concept ” is a theoretical ideal which allows the implementation of objects at this level of abstraction and beyond in a mathematically consistent way. The AXIOM computer algebra system is an implementation of a major part of the Scratchpad Concept. AXIOM (formerly called Scratchpad) is a language with extensible parameterized types and generic operators which is based on the notions of domains and categories [Lambe1], [Jenks-Sutor]. By examining some aspects of the AXIOM system, the Scratchpad Concept will be illustrated. It will be shown how some complex problems in homological algebra were solved through the use of this system. New paradigms are evolving in computer science. There is a thrust towards type