Xingqiang Xiu

• Commutative Gröbner bases have a lot of applications in theory and practice, because they have many nice properties, they are computable, and there exist many efficient improvements of their computations. Non-commutative Gröbner bases also have many useful properties. However, applications of non-commutative Gröbner bases are rarely considered due to high complexity of computations. The purpose of this study was to improve the computation of non-commutative Gröbner bases and investigate the applications of non-commutative Gröbner bases. Gröbner basis theory in free monoid rings was carefully revised and Gröbner bases were precisely characterized in great detail. For the computations of Gröbner bases, the Buchberger Procedure was formulated. Three methods, say interreduction on obstructions, Gebauer-Möller criteria, and detecting redundant generators, were developed for efficiently improving the Buchberger Procedure. Further, the same approach was applied to study Gröbner basis theory in free bimodules over free monoid rings. TheCommutative Gröbner bases have a lot of applications in theory and practice, because they have many nice properties, they are computable, and there exist many efficient improvements of their computations. Non-commutative Gröbner bases also have many useful properties. However, applications of non-commutative Gröbner bases are rarely considered due to high complexity of computations. The purpose of this study was to improve the computation of non-commutative Gröbner bases and investigate the applications of non-commutative Gröbner bases. Gröbner basis theory in free monoid rings was carefully revised and Gröbner bases were precisely characterized in great detail. For the computations of Gröbner bases, the Buchberger Procedure was formulated. Three methods, say interreduction on obstructions, Gebauer-Möller criteria, and detecting redundant generators, were developed for efficiently improving the Buchberger Procedure. Further, the same approach was applied to study Gröbner basis theory in free bimodules over free monoid rings. The Buchberger Procedure was also formulated and improved in this setting. Moreover, J.-C. Faugere's F4 algorithm was generalized to this setting. Finally, many meaningful applications of non-commutative Gröbner bases were developed. Enumerating procedures were proposed to semi-decide some interesting undecidable problems. All the examples in the thesis were computed using the package gbmr of the computer algebra system ApCoCoA. The package was developed by the author. It contains dozens of functions for Gröbner basis computations and many concrete applications. The package gbmr and a collection of interesting examples are available at http://www.apcocoa.org/.…