J. Lyn Miller
Western Kentucky University
This talk summarizes the results of my 1994 doctoral dissertation,
in which methods were presented
for generalizing the theory of Gröbner bases
to four distinct contexts:
First, we outline the results
developed by Robbiano and Sweedler in 1987
and by Sweedler in 1988 to extend
Gröbner basis theory in an intrinsic manner to k-subalgebras of a
polynomial ring k[x1,...,xn]
over a field k
and to their ideals.
We will present additional results that enable us
actually to compute SAGBI-Gröbner bases
(analogs to Gröbner bases for ideals of k-subalgebras) in
k[x1,...,xn]
and also to use them in applications.
Then, we will extend SAGBI and SAGBI-Gröbner theory to the more
general
context of a polynomial ring over a noetherian integral domain,
describing algorithms for computing
these bases and offering some
applications.