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.