The computerisation of algebraic geometry

Abstract

This paper is concerned with the problems of performing computer algebra when the variables involved are related by some algebraic dependencies. It is shown that heuristic or ad hoc treatment of such cases leads rapidly to problems, and the proper mathematical foundations for the treatment of algebraic functions are presented. The formalism leads directly to the requirement for algorithms to find the genus of an algebraic curve, and to discover what function, if any, is associated with a given divisor. These algorithms and the relevant computational techniques are briefly described. In a concluding section the areas where these techniques are required in an integration scheme for algebraic functions are explained.