The polynomial system **F** given by

has the same solutions as

where is any Gröbner basis of **F** with respect to any monomial
ordering.

** Example.** If

then a Gröbner basis system for this polynomial system is (by Maple)

The reader can verify by Maple that every solution of the Gröbner basis is
also a solution of the original system.
It is a little harder to do an independent
check that there are no solutions missed by this method, but one can use the
theory of resultants to find a set of candidate solutions, for example, and if
this is done (sure enough) we find that the solutions are indeed the same.

Tue Mar 12 21:09:19 EST 1996