Computing Gröbner Bases by FGLM Techniques in a Non-commutative Setting

It is well known that the complexity of Gröbner bases computation strongly depends on the term ordering, moreover, elimination orderings often yield a greater complexity. This remark led to the so-called FGLM conversion problem, i.e. given a Gröbner basis w.r.t. a certain term ordering,‖ find a Gröbner basis of the same ideal w.r.t. another term ordering. One of the efficient approaches for solving this problem, in the zero-dimensional case, is the FGLM algorithm (see Faugère et al., 1993). The key ideas of this algorithm were successfully generalized in Marinari et al. (1993) with the objective of computing Gröbner bases of zero-dimensional ideals that are determined by functionals (in the sense that they are kernels of finite sets of linear morphisms from the polynomial ring to the base field). In fact Buchberger and Möller (1982) pioneered the work of FGLM and these algorithms. The main goal of this paper is to generalize the FGLM algorithm to non-commutative polynomial rings.∗∗ Before giving a brief summary of the sections of this paper, let us introduce some familiar notation.