Examples in Search of Algorithms

Introduction Commutativity theorems are part of the study of polynomial identities in noncommutative rings. They are theorems which assert that, under certain conditions, the ring at hand must be commutative. The proofs of theorems of this sort in their general form require the structure theory for non-commutative rings. Instances of these theorems have a strongly computational flavor. They provide interesting test examples for algorithms which use rewrite rules and reduction theory for polynomial rings in non-commuting variables. This paper presents several examples of commutativity theorems with solutions. The solutions were obtained using a reduction process for non-commutative polynomials with integer coefficients. The reduction process blends a treatment of integer coefficients due to Buchberger with handling of non-commutative polynomials due to Mora. Some comparisons are made between automated solutions and solutions “by hand”.