"Factor-SAGBI Bases: a Tool for Computations in Subalgebras of Factor
Algebras"
Patrik Nordbeck.
Department of Mathematics, Lund
Box 118
Lund, S-221 00
Sweden
E-mail:nordbeck@maths.lth.se
We introduce canonical bases for subalgebras of quotients of polynomial rings.
Canonical bases for subalgebras of polynomial rings were introduced by Kapur
and Madlener, and independently by Robbiano and Sweedler. Using the language of
Robbiano and Sweedler and referring to the ``non-quotient'' case as SAGBI bases
theory (Subalgebra Analog to Groebner Bases for Ideals), we consequently call
the canonical bases in our factor algebra setting Factor-SAGBI bases.
SAGBI bases theory is (as the previous parenthesis indicates) strongly
influenced by the theory of Groebner bases, introduced by Bruno Buchberger in
his thesis; in e.g. the paper by Robbiano and Sweedler we find 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 Groebner basis reduction. In the test and
construction of our bases we are forced to consider, besides critical pairs,
one additional type of element.