An Algorithm for Constructing Gröbner and Free Schreier Bases in Free Group Algebras

Abstract

We present an algorithm for computing Gröbner and free canonical Schreier bases for finitely generated one-sided ideals in free group algebras. That is, given a set of n elements of a free group algebra, we construct a canonical Schreier basis of free generators as well as a Gröbner basis for the right ideal generated by the original set. In contrast to Buchberger's algorithm in polynomial rings, at any stage the current number of polynomials does not exceed 2 n and their maximal degree is bounded by d + 2, where d is the maximal degree of the original polynomials. A corollary is that the generalised membership problem for free group algebras is solvable. As a special case we obtain an algorithm similar to the Nielsen-Hall algorithm for constructing free bases for subgroups of free groups. A generalisation of the notion of a Gröbner basis is given by the definition of algebras satisfying constructive division algorithms.