String Rewriting and Gröbner Bases - A General Approach to Monoid and Group Rings

The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semi-Thue system. For certain presentations, including free groups and context-free groups, the existence of finite Gröbner bases for finitely generated right ideals is shown and a procedure to com..

Creative Telescoping for Holonomic Functions

Abstract The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts.