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).