Advances and Trends in the Design and Construction of Algebraic

We compare and contrast several techniques for the implementation ofcomponents of an algebraic manipulation system. On one hand is the mathematicalalgebraic approach which characterizes (for example) IBM's Scratchpad II. On the other hand is the more ad hoc approach which characterizes many other popular systems (for example, Macsyma, Reduce, Maple, and Mathematica). While the algebraic approach has generally positive results, careful examination suggests that there are signi cant remaining problems, especially in the representation and manipulation of analytical, as opposed to algebraic mathematics. We describe some of these problems, and some general approaches for solutions.