The polynomial system F has a solution if and only if the Gröbner basis with respect to any ordering is not the single polynomial .
Example. Suppose . Issuing the commands
with(grobner); gbasis(F, [x, y], tdeg);yields . Hence the theory says there are no solutions to this system (well, you wouldn't expect any since there are three equations in 2 unknowns, and in fact this is easy to verify by hand in this example).