### Lemma 6.2

Let be a complex cubic form. Then **F** is
reduced if and only if:

### Proof

The condition **Q<P** is equivalent to hence 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.