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.Comment: 378+x+I Pages; PhD Thesis (University of Wales, Bangor); Code
available at http://www.dilan4.freeserve.co.uk/maths
