The polynomial system F given by
has the same solutions as
where is any Gröbner basis of F with respect to any monomial ordering.
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.