This special issue of the Journal of Symbolic Computation is a collection of papers arising from the Magma conference that was held at Queen Mary and Westfield College, London, 23-27 August, 1993, and organized by Charles Leedham-Green. The Magma system, developed by John Cannon and associates at Sydney University, is a successor to CAYLEY and, like its predecessor, is geared towards efficient computation within specific, and well-defined, mathematical structures, such as groups, rings, fields, modules, algebras and incidence structures.
To quote from Geddes et al. (1992), there are three recognizable forces at work in the development of symbolic mathematical computation, namely algorithms, systems, and applications. All three themes are well-represented in the papers in this volume and although several of the papers cover material from more than one of the themes, we have attempted to order the contents according to this categorization.
The system papers include detailed descriptions by their authors of the design philosophy, together with a summary of the syntax and scope, of the Magma language. The algorithm and application papers cover computation in such diverse areas as group theory (finite and infinite), number theory, polynomial algebra (including GrΓΆbner bases), Galois theory, lattices and modules. Attention is generally focused on efficient implementation of algorithms (either as stand-alone programs or as part of systems) and performance analysis, in addition to mathematically accurate theoretical descriptions.
With such a broad range of subject matter, the question arises as to just what are the distinguishing features of the areas of mathematics or computer algebra that concern us here. This question would have been easier to answer 10 or 15 years ago, so let us start by casting our minds back to the early 1980s.
In 1982, a conference on "Computational Group Theory" was held in Durham, England, and the proceedings were published in Atkinson (1984). At that time, the area of computational group theory stood out clearly as something distinct from the rest of symbolic computation. The most striking distinguishing feature was the emphasis on manipulating complete structures (principally groups and their subgroups, and character tables) rather than just the elements of these structures. The actual computations that were carried out for nontrivial applications were more often than not exceedingly cpu and memory intensive, and so efficiency of implementation was of paramount importance, whereas generality and portability of software were considered secondary. In fact, the whole field revolved around two or three central families of associated algorithms, most notably Todd-Coxeter methods for handling finitely presented groups, and Schreier-Sims and base/strong-generating set methods for finite permutation groups.
The most significant change since that time is that, even if one wanted to, it would no longer be possible to treat a particular area of computational algebra, such as computational group theory, as an isolated and self-sufficient branch of mathematics. As the