Details:
Title  Noncommutative Involutive Bases  Author(s)  Gareth Evans  Type  PhD Theses  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.  Keywords  13P10, 16Z05 (Primary); 16S15 (Secondary)  Length  389 
File 
 URL 
http://arxiv.org/abs/math.RA/0602140 
Language  English  Year  2005  Month  September  Note  Code available at http://www.dilan4.freeserve.co.uk/maths/  Edition  0  Translation 
No  Refereed 
Yes  Institution 
University of Wales, Bangor  Sponsors  EPSRC; University of Wales, Bangor 
