"Practical Computations with Gr\"obner Bases"

     Daniel Lichtblau
     Wolfram Research, Incorporated
     100 Trade Centre Dr.
     Champaign, IL, 61820
     USA
     E-mail:danl@wolfram.com
	      

Gr\"obner bases have become a pervasive tool in computational mathematics.
In this paper we show several sorts of Gr\"obner basis computations in
Mathematica. We discuss computations of approximate Gr\"obner bases, with
applications to solving systems and implicitization. We also show how to
use the built-in GroebnerBasis function to compute polynomial
greatest-common-divisors over various fields, series inverses, syzygy
modules, and more. Much of this is by now classical; the emphasis is not
on new results but rather on effective use of off-the-shelf technology.
We will point out areas where existing functionality needs enhancement,
and note some directions for future work.