Now, has exactly two real conjugates corresponding to the two
conjugates
and
of
. Since
for all k,
it is easy to see that the conjugate of
corresponding to
is
. Let us denote this
conjugate by
. For
,
has two real
conjugates
and
itself. By definition,
. Since all other conjugates are nonreal, the sign of
the product of all conjugates is determined by that of
, and thus
Since the determinant computed above is just the resultant of and
,
another alternative is to use the coefficient vector of
to represent
and replace the determinant computation by the computation of a resultant at
each step. Or, combining the approaches of § 5.1 and § 5.2, one could compute
using floating point except in ``delicate'' cases.
Experiments showed that the approach of § 5.2 was generally considerably slower than
that of § 5.1.