I take it for granted that you are interested in solving systems of multivariate polynomial equations. Such systems arise in neural networks, chemical reactions, computer-aided design, singular point analysis of algebraic curves, and many others. Much computer algebraic manipulation is in fact simple polynomial system manipulation, which is why it is of intrinsic interest to computer algebra designers. Mathematica incorporates Gröbner bases directly into its simplification routines, for example.

We begin with a vague definition, and after some examples which should
convince you that it is worthwhile to learn about Gröbner bases, I'll give a
real definition.

Tue Mar 12 21:09:19 EST 1996