author = {Katsusuke Nabeshima},
title = {{Comprehensive {G}roebner {B}ases in Various Domains}},
language = {english},
abstract = {This thesis presents algorithms for computing comprehensive Groebner bases and comprehensive Groebner systems in various domains: commutative polynomial rings, rings of differential operators, polynomial rings over a commutative von Neumann regular ring and modules. In both space and time complexity, our new algorithm is much more efficient than other existing algorithms. We define reduced Groebner bases for polynomial rings over a polynomial ring, and introduce algorithms for computing them. There exist some algorithms for computing Groebner bases in polynomial rings over a polynomial ring. However, we cannot obtain the reduced Groebner bases by the algorithms in these rings. We propose a new notion of reduced Groebner bases in these rings. Algorithms for computing Groebner bases in rings of differential operators with coefficients in a polynomial ring, have been proposed in the literature. In this thesis, we present a much more efficient and simpler algorithm than these algorithms by using the relations of two kinds of Groebner bases in rings of differential operators. Moreover, we introduce algorithms for computing their comprehensive Groebner bases. Namely, we describe noncommutative comprehensive Grobner bases in rings of differential operators. Several algorithms are known for computing comprehensive Groebner bases in polynomial rings. However, the extension of comprehensive Groebner bases to modules has not been studied yet. We generalize the theory of comprehensive Groebner bases to the modules. Part of the thesis is an implementation of the presented algorithms in form of a software package for the computer algebra system Risa/Asir.},
number = {07-15},
year = {2007},
month = {April},
note = {PhD thesis},
keywords = {comprehensive Groebner bases, Weyl algebra, modules},
sponsor = {Austrian science foundation (FWF) project P16357-N04, and the SFB F1301},
length = {194},
type = {PhD thesis},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}