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Algorithms for Extending Gr\"obner Bases to Subalgebras and
                          Their Ideals
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J. Lyn Miller
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Date:  July 19th (Friday)\\
Time:  08:30-09:00\\

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Abstract
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This talk summarizes the results of my 1994 doctoral dissertation, in which methods were
presented for generalizing the theory of Gr\"obner bases to four distinct contexts: First, we outline
the results developed by Robbiano and Sweedler in 1987 and by Sweedler in 1988 to extend
Gr\"obner 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-Gr\"obner 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-Gr\"obner 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. 

