On Groebner Bases of (Non)commutative Free Algebras

Abstract

The set of all polynomials in a collection of variables with coefficients in a given field is an important mathematical object, called the polynomial ring. The primary objects of study in this theory are sets of polynomials that contain additional mathematical structure called ideals. The theory of Groebner bases provide a theoretical foundation for answering questions involving ideals. The original algorithm used to produce a Groebner basis was developed in 1976 by Buchberger. It has been implemented in many computer algebra systems. In a paper in 1999, Faugere developed a modification of Buchberger’s algorithm. His algorithm uses row reduction of matrices to perform several steps of the algorithm at once. The goal for the project will be to develop a new implementation of Faugere’s F4 algorithm and to explore new term orders in the noncommutative free algebra as well as applications of Faugere's F4 algorithm to ideals in polynomial rings.