Factor-SAGBI Bases: a Tool for Computations in Subalgebras of Factor Algebras

We introduce canonical bases for subalgebras of quotients of the commutative and non-commutative polynomial ring. A more complete exposition can be found in 4]. Canonical bases for subalgebras of the commutative polynomial ring were introduced by Kapur and Madlener (see 2]), and independently by Robbiano and Sweedler ((5]). Some notes on the non-commutative case can be found in 3]. Using the language of Robbiano and Sweedler, we will refer to these \non-quotient" cases as SAGBI bases theory (Subalgebra Analog to Grr obner Bases for Ideals). In consequence, we will call the canonical bases in our factor algebra setting Factor-SAGBI bases, or simply FS-bases. SAGBI bases theory is (as the previous parenthesis indicates) strongly innuenced by the theory of Grr obner bases, introduced by Bruno Buchberger in his thesis 1]; in e.g. 5] we nd the notion of (subalgebra) reduction, the characterization (test) theorem using critical pairs (generalized S-polynomials), and the completion procedure of constructing bases. To make the theory work in our factor algebra setting, we need just complete the SAGBI theory at a few points. We try, as far as possible, to work in the normal complements of the ideals we factor out, so e.g. our subalgebra reduction also includes the usual Grr obner basis reduction (called normalization below). In the test and construction of our bases we are forced to consider, besides critical pairs, one additional type of element. Some problems concerning subalgebras of factor algebras, e.g. subalgebra membership, can be reduced to Grr obner basis problems. In the commutative case, FS-bases, like SAGBI bases, diier from Grr obner bases at one essential point; they may be innnite even for ni-tely generated subalgebras. This implies that the Factor-SAGBI approach is not necessarily algorithmic 1. However, it is easy to provide examples where the FS-basis computation is almost free whereas it is impossible, from the practical point of view, to apply the Grr obner bases method mentioned above. Passing to the non-commutative case, Grr obner bases are no longer in general nite, and we can easily nd examples where Factor-SAGBI theory solves problems that would not be algorithmic using the Grr obner bases method. Non-commutative FS-bases have also shown to be applicable for solving systems of non-commutative polynomial equations, in an approach under development by Victor Ufnarovski. The author expresses his thanks to Victor Ufnarovski for helpful discussions. Notations Since most of the theory will be the same for …