Computing Grr Obner Bases by Fglm Techniques in a Noncommutative Setting

A generalization of the FGLM technique is given to compute Grr obner bases for two-sided ideals of free nitely generated algebras. Specializations of this algorithm are presented for the cases in which the ideal is determined by either functionals or monoid (group) presentations. Generalizations are discussed in order to compute G-bases on (twisted) semigroup rings. It is well known that the complexity of Grr obner bases computation strongly depends on the term ordering, moreover, elimination orderings often yield a greater complexity. This remark led to the so called FGLM convertion problem, i.e., given a Grr obner basis with respect to a certain term ordering y , nd a Grr obner basis of the same ideal with respect to another term ordering. One of the eecient approaches for solving this problem, in the zero-dimensional case, is the FGLM algorithm (see FGLM]). The key ideas of this algorithm were successfully generalized in MMM] with the objective of computing Grr obner bases of zero-dimensional ideals that are determined by functionals (in the sense that they are kernels of nite sets of linear morphisms from the polynomial ring to the base eld). In fact, the work pioneer of FGLM and these algorithms was BM]. The main goal of this paper is to generalize FGLM algorithm to noncommutative polynomial rings z. Before giving a brief summary of the sections of this paper, let us introduce some familiar notation: y Usually, it is a total degree ordering, where computing complexity is lower. z The theory presented here in the case of two-sided ideals can be generalized to left-modules of non-commutative polynomial rings (see Al]).