Noncommutative Involutive Bases

TL;DR

This paper extends the theory of involutive bases to noncommutative polynomial rings, providing algorithms for noncommutative involutive divisions, basis computation, and a noncommutative Groebner Walk, advancing symbolic computation methods.

Contribution

It introduces new algorithms for noncommutative involutive divisions and bases, generalizing existing commutative methods and providing a framework for noncommutative Groebner basis computation.

Findings

Algorithms for various noncommutative involutive divisions are developed.

A noncommutative Involutive Basis algorithm is proposed, yielding Groebner Bases.

A noncommutative Groebner Walk algorithm is introduced for monomial order conversions.

Abstract

The theory of Groebner Bases originated in the work of Buchberger and is now considered to be one of the most important and useful areas of symbolic computation. A great deal of effort has been put into improving Buchberger's algorithm for computing a Groebner Basis, and indeed in finding alternative methods of computing Groebner Bases. Two of these methods include the Groebner Walk method and the computation of Involutive Bases. By the mid 1980's, Buchberger's work had been generalised for noncommutative polynomial rings by Bergman and Mora. This thesis provides the corresponding generalisation for Involutive Bases and (to a lesser extent) the Groebner Walk, with the main results being as follows. (1) Algorithms for several new noncommutative involutive divisions are given, including strong; weak; global and local divisions. (2) An algorithm for computing a noncommutative Involutive Basis is given. When used with one of the aforementioned involutive divisions, it is shown that this algorithm returns a noncommutative Groebner Basis on termination. (3) An algorithm for a noncommutative Groebner Walk is given, in the case of conversion between two harmonious monomial orderings. It is shown that this algorithm generalises to give an algorithm for performing a noncommutative Involutive Walk, again in the case of conversion between two harmonious monomial orderings. (4) Two new properties of commutative involutive divisions are introduced (stability and extendibility), respectively ensuring the termination of the Involutive Basis algorithm and the applicability (under certain conditions) of homogeneous methods of computing Involutive Bases.