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Recipe 0---Main property

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.

Robert Corless
Tue Mar 12 21:09:19 EST 1996