Letbe a complex cubic form. Then F is reduced if and only if:
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The condition Q<P is equivalent tohence to
since a>0. Since F has only one real root (and again a>0), this is equivalent to
which gives
. Similarly, the condition -Q<P is equivalent to
hence to
, so
which gives
. Finally, the condition P<R is equivalent to
, and this is equivalent to
where
is the resultant,
and
. An immediate computation shows that
, and this gives the last unproved condition.