### Abstract:

Groebner-Shirshov bases, introduced independently by Shirshov in 1962 and Buchberger in 1965, are powerful computational tools in mathematics, science, engineering, and computer science. This thesis focuses on the theories, algorithms, and applications of Groebner-Shirshov bases for two classes of noncommutative algebras: differential difference algebras and skew solvable polynomial rings.
This thesis consists of three manuscripts (Chapters 2--4), an introductory chapter (Chapter 1) and a concluding chapter (Chapter 5).
In Chapter 1, we introduce the background and the goals of the thesis.
In Chapter 2, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We find lower and upper bounds of the Gelfand-Kirillov dimension of a differential difference algebra under some conditions. We also give examples to demonstrate that our bounds are sharp.
In Chapter 3, we generalize the Groebner-Shirshov basis theory to differential difference algebras with respect to any left admissible ordering and develop the Groebner-Shirshov basis theory of finitely generated free modules over differential difference algebras. By using the theory we develop, we present an algorithm to compute the Gelfand-Kirillov dimensions of finitely generated modules over differential difference algebras.
In Chapter 4, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Groebner-Shirshov bases in skew solvable polynomial rings over fields. Our algorithm can detect redundant reductions and therefore it is more efficient than the traditional Buchberger algorithm.
Finally, in Chapter 5, we summarize our results and propose possible future work.